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Math Problems
Calculus
Evaluate definite integrals using the chain rule
How do you write
0.
1
‾
0.\overline{1}
0.
1
as a fraction?
\newline
Choices:
\newline
(A)
1
6
\frac{1}{6}
6
1
\newline
(B)
1
8
\frac{1}{8}
8
1
\newline
(C)
1
9
\frac{1}{9}
9
1
\newline
(D)
1
3
\frac{1}{3}
3
1
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Evaluate the definite integral:
y
=
∫
0
2
2
x
e
x
d
x
y=\int_{0}^{2} 2xe^{x}\,dx
y
=
∫
0
2
2
x
e
x
d
x
Get tutor help
Simplify the following algebraic fraction:
\newline
2
x
2
−
3
x
−
5
x
2
+
6
x
+
5
\frac{2x^{2}-3x-5}{x^{2}+6x+5}
x
2
+
6
x
+
5
2
x
2
−
3
x
−
5
Get tutor help
Evaluate
∫
1
x
2
+
6
x
+
13
d
x
\int\frac{1}{x^{2}+6x+13}\,dx
∫
x
2
+
6
x
+
13
1
d
x
\newline
(A)
ln
∣
(
x
+
3
)
2
+
4
∣
+
C
\ln|\left(x+3\right)^{2}+4|+C
ln
∣
(
x
+
3
)
2
+
4∣
+
C
\newline
(B)
−
1
2
tan
−
1
(
x
+
3
2
)
+
C
-\frac{1}{2}\tan^{-1}\left(\frac{x+3}{2}\right)+C
−
2
1
tan
−
1
(
2
x
+
3
)
+
C
\newline
(C)
−
(
1
x
2
+
6
x
+
13
)
−
2
+
C
-\left(\frac{1}{x^{2}+6x+13}\right)^{-2}+C
−
(
x
2
+
6
x
+
13
1
)
−
2
+
C
\newline
(D)
1
2
tan
−
1
(
x
+
3
2
)
+
C
\frac{1}{2}\tan^{-1}\left(\frac{x+3}{2}\right)+C
2
1
tan
−
1
(
2
x
+
3
)
+
C
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∫
π
6
π
3
(
1
cos
2
x
−
1
sin
2
x
)
d
x
\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\left(\frac{1}{\cos ^{2} x}-\frac{1}{\sin ^{2} x}\right) d x
∫
6
π
3
π
(
cos
2
x
1
−
sin
2
x
1
)
d
x
\newline
0
0
0
,
5
5
5
\newline
0
0
0
Get tutor help
Solve:
∫
0
1
x
d
x
x
4
+
1
=
\int_{0}^{1}\frac{x\,dx}{\sqrt{x^{4}+1}}=
∫
0
1
x
4
+
1
x
d
x
=
Get tutor help
∫
1
e
−
x
−
1
d
x
\int \frac{1}{e^{-x}-1} d x
∫
e
−
x
−
1
1
d
x
Get tutor help
∫
x
2
d
x
x
2
−
4
\int \frac{x^{2} d x}{x^{2}-4}
∫
x
2
−
4
x
2
d
x
=
Get tutor help
lim
(
x
,
y
)
→
(
0
,
0
)
x
2
+
y
2
x
2
+
y
2
+
1
−
1
\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{\sqrt{x^{2}+y^{2}+1}-1}
lim
(
x
,
y
)
→
(
0
,
0
)
x
2
+
y
2
+
1
−
1
x
2
+
y
2
Get tutor help
If
f
(
x
)
=
(
x
2
+
1
)
3
f(x)=\left(x^{2}+1\right)^{3}
f
(
x
)
=
(
x
2
+
1
)
3
, what is
lim
x
→
−
1
f
(
x
)
−
f
(
−
1
)
x
+
1
?
\lim _{x \rightarrow-1} \frac{f(x)-f(-1)}{x+1} ?
lim
x
→
−
1
x
+
1
f
(
x
)
−
f
(
−
1
)
?
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∫
−
π
6
π
3
−
sin
x
cos
2
x
d
x
\int_{-\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{-\sin x}{\cos ^{2} x} d x
∫
−
6
π
3
π
cos
2
x
−
sin
x
d
x
\newline
Get tutor help
r
(
x
)
=
∫
−
2
x
2
t
e
4
t
2
d
h
r(x)=\int_{-2}^{x}2te^{4t^{2}}\,dh
r
(
x
)
=
∫
−
2
x
2
t
e
4
t
2
d
h
Get tutor help
Select all the expressions that are equivalent to
6
−
4
×
6
−
4
6^{-4} \times 6^{-4}
6
−
4
×
6
−
4
.
\newline
Multi-select Choices:
\newline
(A)
1
6
16
\frac{1}{6^{16}}
6
16
1
\newline
(B)
6
16
6^{16}
6
16
\newline
(C)
1
6
−
8
\frac{1}{6^{-8}}
6
−
8
1
\newline
(D)
6
−
8
6^{-8}
6
−
8
Get tutor help
Evaluate the indefinite integral:
∫
(
1
+
x
/
2
)
8
d
x
\int(1+x/2)^{8}\,dx
∫
(
1
+
x
/2
)
8
d
x
Get tutor help
Find
∫
1
−
x
2
−
4
x
+
21
d
x
\int \frac{1}{\sqrt{-x^{2}-4 x+21}} d x
∫
−
x
2
−
4
x
+
21
1
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
5
arctan
(
x
+
2
5
)
+
C
\frac{1}{5} \arctan \left(\frac{x+2}{5}\right)+C
5
1
arctan
(
5
x
+
2
)
+
C
\newline
(B)
1
5
arcsin
(
x
+
2
5
)
+
C
\frac{1}{5} \arcsin \left(\frac{x+2}{5}\right)+C
5
1
arcsin
(
5
x
+
2
)
+
C
\newline
(c)
arctan
(
x
+
2
5
)
+
C
\arctan \left(\frac{x+2}{5}\right)+C
arctan
(
5
x
+
2
)
+
C
\newline
(D)
arcsin
(
x
+
2
5
)
+
C
\arcsin \left(\frac{x+2}{5}\right)+C
arcsin
(
5
x
+
2
)
+
C
Get tutor help
Find
∫
1
−
x
2
−
6
x
+
40
d
x
\int \frac{1}{\sqrt{-x^{2}-6 x+40}} d x
∫
−
x
2
−
6
x
+
40
1
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
14
arctan
(
x
+
3
7
)
+
C
\frac{1}{14} \arctan \left(\frac{x+3}{7}\right)+C
14
1
arctan
(
7
x
+
3
)
+
C
\newline
(B)
1
14
arcsin
(
x
+
3
7
)
+
C
\frac{1}{14} \arcsin \left(\frac{x+3}{7}\right)+C
14
1
arcsin
(
7
x
+
3
)
+
C
\newline
(c)
arctan
(
x
+
3
7
)
+
C
\arctan \left(\frac{x+3}{7}\right)+C
arctan
(
7
x
+
3
)
+
C
\newline
()
arcsin
(
x
+
3
7
)
+
C
\arcsin \left(\frac{x+3}{7}\right)+C
arcsin
(
7
x
+
3
)
+
C
Get tutor help
Evaluate the integral
∫
3
x
+
4
x
+
3
d
x
\int \frac{3 x+4}{x+3} d x
∫
x
+
3
3
x
+
4
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
=
9
x
−
4
ln
∣
x
+
3
∣
+
C
=9 x-4 \ln |x+3|+C
=
9
x
−
4
ln
∣
x
+
3∣
+
C
\newline
(B)
=
3
x
−
5
ln
∣
x
+
3
∣
+
C
=3 x-5 \ln |x+3|+C
=
3
x
−
5
ln
∣
x
+
3∣
+
C
\newline
(C)
=
3
x
+
5
ln
∣
x
+
3
∣
+
C
=3 x+5 \ln |x+3|+C
=
3
x
+
5
ln
∣
x
+
3∣
+
C
\newline
(D)
=
9
x
−
5
ln
∣
x
+
3
∣
+
C
=9 x-5 \ln |x+3|+C
=
9
x
−
5
ln
∣
x
+
3∣
+
C
Get tutor help
Evaluate the integral
∫
x
−
1
2
x
+
4
d
x
\int \frac{x-1}{2 x+4} d x
∫
2
x
+
4
x
−
1
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
2
x
−
1
2
ln
∣
x
+
2
∣
+
C
\frac{1}{2} x-\frac{1}{2} \ln |x+2|+C
2
1
x
−
2
1
ln
∣
x
+
2∣
+
C
\newline
(B)
1
2
x
−
ln
∣
x
+
2
∣
+
C
\frac{1}{2} x-\ln |x+2|+C
2
1
x
−
ln
∣
x
+
2∣
+
C
\newline
(C)
1
2
x
−
3
2
ln
∣
x
+
2
∣
+
C
\frac{1}{2} x-\frac{3}{2} \ln |x+2|+C
2
1
x
−
2
3
ln
∣
x
+
2∣
+
C
\newline
(D)
1
2
x
−
2
ln
∣
x
+
2
∣
+
C
\frac{1}{2} x-2 \ln |x+2|+C
2
1
x
−
2
ln
∣
x
+
2∣
+
C
Get tutor help
Evaluate
∫
x
6
+
2
x
4
+
6
x
−
9
x
3
+
3
d
x
\int \frac{x^{6}+2 x^{4}+6 x-9}{x^{3}+3} d x
∫
x
3
+
3
x
6
+
2
x
4
+
6
x
−
9
d
x
\newline
Choose
1
1
1
answer:
\newline
(A)
x
4
4
+
x
2
−
3
x
+
C
\frac{x^{4}}{4}+x^{2}-3 x+C
4
x
4
+
x
2
−
3
x
+
C
\newline
(B)
x
4
4
+
x
2
−
9
x
+
C
\frac{x^{4}}{4}+x^{2}-9 x+C
4
x
4
+
x
2
−
9
x
+
C
\newline
(C)
x
4
4
+
3
x
2
+
3
x
+
C
\frac{x^{4}}{4}+3 x^{2}+3 x+C
4
x
4
+
3
x
2
+
3
x
+
C
\newline
(D)
x
4
4
+
x
2
+
3
x
+
C
\frac{x^{4}}{4}+x^{2}+3 x+C
4
x
4
+
x
2
+
3
x
+
C
Get tutor help
Evaluate
∫
x
3
−
1
x
+
2
d
x
\int \frac{x^{3}-1}{x+2} d x
∫
x
+
2
x
3
−
1
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
x
3
3
−
x
2
+
4
x
−
7
ln
∣
x
+
2
∣
+
C
\frac{x^{3}}{3}-x^{2}+4 x-7 \ln |x+2|+C
3
x
3
−
x
2
+
4
x
−
7
ln
∣
x
+
2∣
+
C
\newline
(B)
x
3
3
−
2
x
2
+
4
x
−
7
ln
∣
x
+
2
∣
+
C
\frac{x^{3}}{3}-2 x^{2}+4 x-7 \ln |x+2|+C
3
x
3
−
2
x
2
+
4
x
−
7
ln
∣
x
+
2∣
+
C
\newline
(C)
x
3
3
−
x
2
+
4
x
−
9
ln
∣
x
+
2
∣
+
C
\frac{x^{3}}{3}-x^{2}+4 x-9 \ln |x+2|+C
3
x
3
−
x
2
+
4
x
−
9
ln
∣
x
+
2∣
+
C
\newline
(D)
x
3
3
−
2
x
2
+
4
x
−
9
ln
∣
x
+
2
∣
+
C
\frac{x^{3}}{3}-2 x^{2}+4 x-9 \ln |x+2|+C
3
x
3
−
2
x
2
+
4
x
−
9
ln
∣
x
+
2∣
+
C
Get tutor help
Evaluate the integral
∫
x
−
5
−
2
x
+
2
d
x
\int \frac{x-5}{-2 x+2} d x
∫
−
2
x
+
2
x
−
5
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
2
x
−
2
ln
∣
x
−
1
∣
+
C
\frac{1}{2} x-2 \ln |x-1|+C
2
1
x
−
2
ln
∣
x
−
1∣
+
C
\newline
(B)
1
2
x
+
2
ln
∣
x
−
1
∣
+
C
\frac{1}{2} x+2 \ln |x-1|+C
2
1
x
+
2
ln
∣
x
−
1∣
+
C
\newline
(C)
−
1
2
x
−
2
ln
∣
x
−
1
∣
+
C
-\frac{1}{2} x-2 \ln |x-1|+C
−
2
1
x
−
2
ln
∣
x
−
1∣
+
C
\newline
(D)
−
1
2
x
+
2
ln
∣
x
−
1
∣
+
C
-\frac{1}{2} x+2 \ln |x-1|+C
−
2
1
x
+
2
ln
∣
x
−
1∣
+
C
Get tutor help
Evaluate the integral
∫
2
x
+
1
x
+
2
d
x
\int \frac{2 x+1}{x+2} d x
∫
x
+
2
2
x
+
1
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
2
x
−
3
ln
∣
x
+
2
∣
+
C
2 x-3 \ln |x+2|+C
2
x
−
3
ln
∣
x
+
2∣
+
C
\newline
(B)
ln
∣
x
+
2
∣
+
C
\ln |x+2|+C
ln
∣
x
+
2∣
+
C
\newline
(C)
x
+
ln
∣
2
x
+
1
∣
+
C
x+\ln |2 x+1|+C
x
+
ln
∣2
x
+
1∣
+
C
\newline
(D)
2
x
−
ln
∣
x
+
2
∣
+
C
2 x-\ln |x+2|+C
2
x
−
ln
∣
x
+
2∣
+
C
Get tutor help
Evaluate
∫
2
x
3
+
7
x
2
+
2
x
+
9
2
x
+
3
d
x
\int \frac{2 x^{3}+7 x^{2}+2 x+9}{2 x+3} d x
∫
2
x
+
3
2
x
3
+
7
x
2
+
2
x
+
9
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
x
3
3
+
x
2
−
2
x
+
15
ln
∣
2
x
+
3
∣
2
+
C
\frac{x^{3}}{3}+x^{2}-2 x+\frac{15 \ln |2 x+3|}{2}+C
3
x
3
+
x
2
−
2
x
+
2
15
l
n
∣2
x
+
3∣
+
C
\newline
(B)
x
3
3
+
x
2
−
2
x
+
5
ln
∣
2
x
+
3
∣
2
+
C
\frac{x^{3}}{3}+x^{2}-2 x+\frac{5 \ln |2 x+3|}{2}+C
3
x
3
+
x
2
−
2
x
+
2
5
l
n
∣2
x
+
3∣
+
C
\newline
(C)
x
3
3
+
x
2
2
−
2
x
+
15
ln
∣
2
x
+
3
∣
2
+
C
\frac{x^{3}}{3}+\frac{x^{2}}{2}-2 x+\frac{15 \ln |2 x+3|}{2}+C
3
x
3
+
2
x
2
−
2
x
+
2
15
l
n
∣2
x
+
3∣
+
C
\newline
(D)
x
3
3
+
x
2
2
−
2
x
+
5
ln
∣
2
x
+
3
∣
2
+
C
\frac{x^{3}}{3}+\frac{x^{2}}{2}-2 x+\frac{5 \ln |2 x+3|}{2}+C
3
x
3
+
2
x
2
−
2
x
+
2
5
l
n
∣2
x
+
3∣
+
C
Get tutor help
Evaluate
∫
x
3
+
x
x
+
1
d
x
\int \frac{x^{3}+x}{x+1} d x
∫
x
+
1
x
3
+
x
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
x
3
3
−
x
2
2
+
2
x
−
2
ln
∣
x
+
1
∣
+
C
\frac{x^{3}}{3}-\frac{x^{2}}{2}+2 x-2 \ln |x+1|+C
3
x
3
−
2
x
2
+
2
x
−
2
ln
∣
x
+
1∣
+
C
\newline
(B)
x
3
3
+
2
x
−
2
ln
∣
x
+
1
∣
+
C
\frac{x^{3}}{3}+2 x-2 \ln |x+1|+C
3
x
3
+
2
x
−
2
ln
∣
x
+
1∣
+
C
\newline
(C)
x
3
3
−
x
2
2
+
2
x
+
2
ln
∣
x
+
1
∣
+
C
\frac{x^{3}}{3}-\frac{x^{2}}{2}+2 x+2 \ln |x+1|+C
3
x
3
−
2
x
2
+
2
x
+
2
ln
∣
x
+
1∣
+
C
\newline
(D)
x
3
3
+
x
2
+
2
x
+
2
ln
∣
x
+
1
∣
+
C
\frac{x^{3}}{3}+x^{2}+2 x+2 \ln |x+1|+C
3
x
3
+
x
2
+
2
x
+
2
ln
∣
x
+
1∣
+
C
Get tutor help
Evaluate the integral
∫
x
−
2
3
x
+
6
d
x
\int \frac{x-2}{3 x+6} d x
∫
3
x
+
6
x
−
2
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
2
x
−
4
ln
∣
x
+
2
∣
3
+
C
\frac{1}{2} x-\frac{4 \ln |x+2|}{3}+C
2
1
x
−
3
4
l
n
∣
x
+
2∣
+
C
\newline
(B)
1
2
x
+
4
ln
∣
x
+
2
∣
3
+
C
\frac{1}{2} x+\frac{4 \ln |x+2|}{3}+C
2
1
x
+
3
4
l
n
∣
x
+
2∣
+
C
\newline
(C)
1
3
x
−
4
ln
∣
x
+
2
∣
3
+
C
\frac{1}{3} x-\frac{4 \ln |x+2|}{3}+C
3
1
x
−
3
4
l
n
∣
x
+
2∣
+
C
\newline
(D)
1
3
x
+
4
ln
∣
x
+
2
∣
3
+
C
\frac{1}{3} x+\frac{4 \ln |x+2|}{3}+C
3
1
x
+
3
4
l
n
∣
x
+
2∣
+
C
Get tutor help
Evaluate the integral
∫
x
+
4
2
x
+
6
d
x
\int \frac{x+4}{2 x+6} d x
∫
2
x
+
6
x
+
4
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
2
x
−
ln
∣
2
x
+
6
∣
2
+
C
2 x-\frac{\ln |2 x+6|}{2}+C
2
x
−
2
l
n
∣2
x
+
6∣
+
C
\newline
(B)
1
2
x
−
ln
∣
2
x
+
6
∣
2
+
C
\frac{1}{2} x-\frac{\ln |2 x+6|}{2}+C
2
1
x
−
2
l
n
∣2
x
+
6∣
+
C
\newline
(C)
2
x
+
ln
∣
2
x
+
6
∣
+
C
2 x+\ln |2 x+6|+C
2
x
+
ln
∣2
x
+
6∣
+
C
\newline
(D)
1
2
x
+
ln
∣
2
x
+
6
∣
2
+
C
\frac{1}{2} x+\frac{\ln |2 x+6|}{2}+C
2
1
x
+
2
l
n
∣2
x
+
6∣
+
C
Get tutor help
Find
∫
1
−
x
2
+
12
x
−
32
d
x
\int \frac{1}{\sqrt{-x^{2}+12 x-32}} d x
∫
−
x
2
+
12
x
−
32
1
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
arcsin
(
x
−
6
2
)
+
C
\arcsin \left(\frac{x-6}{2}\right)+C
arcsin
(
2
x
−
6
)
+
C
\newline
(B)
1
12
arctan
(
x
−
6
2
)
+
C
\frac{1}{12} \arctan \left(\frac{x-6}{2}\right)+C
12
1
arctan
(
2
x
−
6
)
+
C
\newline
(C)
1
12
arcsin
(
x
−
6
2
)
+
C
\frac{1}{12} \arcsin \left(\frac{x-6}{2}\right)+C
12
1
arcsin
(
2
x
−
6
)
+
C
\newline
(D)
arctan
(
x
−
6
2
)
+
C
\arctan \left(\frac{x-6}{2}\right)+C
arctan
(
2
x
−
6
)
+
C
Get tutor help
Find
∫
1
x
2
−
6
x
+
13
d
x
\int \frac{1}{x^{2}-6 x+13} d x
∫
x
2
−
6
x
+
13
1
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
2
arcsin
(
x
−
3
2
)
+
C
\frac{1}{2} \arcsin \left(\frac{x-3}{2}\right)+C
2
1
arcsin
(
2
x
−
3
)
+
C
\newline
(B)
1
4
arcsin
(
x
−
3
2
)
+
C
\frac{1}{4} \arcsin \left(\frac{x-3}{2}\right)+C
4
1
arcsin
(
2
x
−
3
)
+
C
\newline
(C)
1
2
arctan
(
x
−
3
2
)
+
C
\frac{1}{2} \arctan \left(\frac{x-3}{2}\right)+C
2
1
arctan
(
2
x
−
3
)
+
C
\newline
(D)
1
4
arctan
(
x
−
3
2
)
+
C
\frac{1}{4} \arctan \left(\frac{x-3}{2}\right)+C
4
1
arctan
(
2
x
−
3
)
+
C
Get tutor help
Find
∫
1
x
2
+
8
x
+
52
d
x
\int \frac{1}{x^{2}+8 x+52} d x
∫
x
2
+
8
x
+
52
1
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
6
arcsin
(
x
+
4
6
)
+
C
\frac{1}{6} \arcsin \left(\frac{x+4}{6}\right)+C
6
1
arcsin
(
6
x
+
4
)
+
C
\newline
(B)
arctan
(
x
+
4
6
)
+
C
\arctan \left(\frac{x+4}{6}\right)+C
arctan
(
6
x
+
4
)
+
C
\newline
(C)
1
6
arctan
(
x
+
4
6
)
+
C
\frac{1}{6} \arctan \left(\frac{x+4}{6}\right)+C
6
1
arctan
(
6
x
+
4
)
+
C
\newline
(D)
arcsin
(
x
+
4
6
)
+
C
\arcsin \left(\frac{x+4}{6}\right)+C
arcsin
(
6
x
+
4
)
+
C
Get tutor help
Find
∫
1
3
x
2
+
6
x
+
78
d
x
\int \frac{1}{3 x^{2}+6 x+78} d x
∫
3
x
2
+
6
x
+
78
1
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
arctan
(
x
+
1
5
)
+
C
\arctan \left(\frac{x+1}{5}\right)+C
arctan
(
5
x
+
1
)
+
C
\newline
(B)
arcsin
(
x
+
1
5
)
+
C
\arcsin \left(\frac{x+1}{5}\right)+C
arcsin
(
5
x
+
1
)
+
C
\newline
(c)
1
15
arctan
(
x
+
1
5
)
+
C
\frac{1}{15} \arctan \left(\frac{x+1}{5}\right)+C
15
1
arctan
(
5
x
+
1
)
+
C
\newline
(D)
1
15
arcsin
(
x
+
1
5
)
+
C
\frac{1}{15} \arcsin \left(\frac{x+1}{5}\right)+C
15
1
arcsin
(
5
x
+
1
)
+
C
Get tutor help
Find
∫
1
−
x
2
+
10
x
+
11
d
x
\int \frac{1}{\sqrt{-x^{2}+10 x+11}} d x
∫
−
x
2
+
10
x
+
11
1
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
6
arctan
(
x
−
5
6
)
+
C
\frac{1}{6} \arctan \left(\frac{x-5}{6}\right)+C
6
1
arctan
(
6
x
−
5
)
+
C
\newline
(B)
1
6
arcsin
(
x
−
5
6
)
+
C
\frac{1}{6} \arcsin \left(\frac{x-5}{6}\right)+C
6
1
arcsin
(
6
x
−
5
)
+
C
\newline
(C)
arcsin
(
x
−
5
6
)
+
C
\arcsin \left(\frac{x-5}{6}\right)+C
arcsin
(
6
x
−
5
)
+
C
\newline
(D)
arctan
(
x
−
5
6
)
+
C
\arctan \left(\frac{x-5}{6}\right)+C
arctan
(
6
x
−
5
)
+
C
Get tutor help
Find
∫
1
x
2
−
14
x
+
58
d
x
\int \frac{1}{x^{2}-14 x+58} d x
∫
x
2
−
14
x
+
58
1
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
21
arcsin
(
x
−
7
3
)
+
C
\frac{1}{21} \arcsin \left(\frac{x-7}{3}\right)+C
21
1
arcsin
(
3
x
−
7
)
+
C
\newline
(B)
1
21
arctan
(
x
−
7
3
)
+
C
\frac{1}{21} \arctan \left(\frac{x-7}{3}\right)+C
21
1
arctan
(
3
x
−
7
)
+
C
\newline
(C)
1
3
arctan
(
x
−
7
3
)
+
C
\frac{1}{3} \arctan \left(\frac{x-7}{3}\right)+C
3
1
arctan
(
3
x
−
7
)
+
C
\newline
(D)
1
3
arcsin
(
x
−
7
3
)
+
C
\frac{1}{3} \arcsin \left(\frac{x-7}{3}\right)+C
3
1
arcsin
(
3
x
−
7
)
+
C
Get tutor help
Find
∫
1
−
x
2
−
14
x
−
48
d
x
\int \frac{1}{\sqrt{-x^{2}-14 x-48}} d x
∫
−
x
2
−
14
x
−
48
1
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
2
arctan
(
x
+
7
2
)
+
C
\frac{1}{2} \arctan \left(\frac{x+7}{2}\right)+C
2
1
arctan
(
2
x
+
7
)
+
C
\newline
(B)
arcsin
(
x
+
7
)
+
C
\arcsin (x+7)+C
arcsin
(
x
+
7
)
+
C
\newline
(C)
arctan
(
x
+
7
)
+
C
\arctan (x+7)+C
arctan
(
x
+
7
)
+
C
\newline
(D)
1
2
arcsin
(
x
+
7
2
)
+
C
\frac{1}{2} \arcsin \left(\frac{x+7}{2}\right)+C
2
1
arcsin
(
2
x
+
7
)
+
C
Get tutor help
Find
∫
1
x
2
+
10
x
+
41
d
x
\int \frac{1}{x^{2}+10 x+41} d x
∫
x
2
+
10
x
+
41
1
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
4
arctan
(
x
+
5
4
)
+
C
\frac{1}{4} \arctan \left(\frac{x+5}{4}\right)+C
4
1
arctan
(
4
x
+
5
)
+
C
\newline
(B)
1
4
arcsin
(
x
+
5
4
)
+
C
\frac{1}{4} \arcsin \left(\frac{x+5}{4}\right)+C
4
1
arcsin
(
4
x
+
5
)
+
C
\newline
(C)
1
20
arcsin
(
x
+
5
4
)
+
C
\frac{1}{20} \arcsin \left(\frac{x+5}{4}\right)+C
20
1
arcsin
(
4
x
+
5
)
+
C
\newline
(D)
1
20
arctan
(
x
+
5
4
)
+
C
\frac{1}{20} \arctan \left(\frac{x+5}{4}\right)+C
20
1
arctan
(
4
x
+
5
)
+
C
Get tutor help
g
(
x
)
=
∫
0
x
10
t
+
1
d
t
g(x)=\int_{0}^{x} \sqrt{10 t+1} d t
g
(
x
)
=
∫
0
x
10
t
+
1
d
t
\newline
g
′
(
8
)
=
g^{\prime}(8)=
g
′
(
8
)
=
Get tutor help
g
(
x
)
=
∫
1
x
2
t
+
7
d
t
g(x)=\int_{1}^{x} \sqrt{2 t+7} d t
g
(
x
)
=
∫
1
x
2
t
+
7
d
t
\newline
g
′
(
9
)
=
g^{\prime}(9)=
g
′
(
9
)
=
Get tutor help
g
(
x
)
=
∫
1
x
19
−
t
d
t
g(x)=\int_{1}^{x} \sqrt{19-t} d t
g
(
x
)
=
∫
1
x
19
−
t
d
t
\newline
g
′
(
3
)
=
g^{\prime}(3)=
g
′
(
3
)
=
Get tutor help
g
(
x
)
=
∫
−
1
x
(
8
−
t
)
d
t
g(x)=\int_{-1}^{x}(8-t) d t
g
(
x
)
=
∫
−
1
x
(
8
−
t
)
d
t
\newline
g
′
(
1
)
=
g^{\prime}(1)=
g
′
(
1
)
=
Get tutor help
g
(
x
)
=
∫
2
x
1
1
+
t
3
d
t
g(x)=\int_{2}^{x} \frac{1}{1+t^{3}} d t
g
(
x
)
=
∫
2
x
1
+
t
3
1
d
t
\newline
g
′
(
2
)
=
g^{\prime}(2)=
g
′
(
2
)
=
\newline
Choose
1
1
1
answer:
\newline
(A)
−
7
18
\frac{-7}{18}
18
−
7
\newline
(B)
0
0
0
\newline
(C)
1
9
\frac{1}{9}
9
1
\newline
(D)
1
3
\frac{1}{3}
3
1
\newline
(E) None of these
Get tutor help
g
(
x
)
=
∫
−
10
x
(
10
−
3
t
)
d
t
g
′
(
−
4
)
=
\begin{array}{l}g(x)=\int_{-10}^{x}(10-3 t) d t \\ g^{\prime}(-4)=\end{array}
g
(
x
)
=
∫
−
10
x
(
10
−
3
t
)
d
t
g
′
(
−
4
)
=
Get tutor help
F
(
x
)
=
∫
0
x
t
2
d
t
F
′
(
x
)
=
\begin{array}{l}F(x)=\int_{0}^{\sqrt{x}} t^{2} d t \\ F^{\prime}(x)=\end{array}
F
(
x
)
=
∫
0
x
t
2
d
t
F
′
(
x
)
=
Get tutor help
Evaluate the integral:
\newline
∫
e
2
x
−
1
e
2
x
+
3
d
x
\int\frac{e^{2x}-1}{e^{2x}+3}dx
∫
e
2
x
+
3
e
2
x
−
1
d
x
Get tutor help
Express
1
−
tan
2
θ
1
+
tan
2
θ
\frac{1-\tan^{2}\theta}{1+\tan^{2}\theta}
1
+
t
a
n
2
θ
1
−
t
a
n
2
θ
in terms of
sin
θ
\sin \theta
sin
θ
.
Get tutor help
∫
0
π
sin
θ
+
sin
θ
tan
2
θ
sec
2
θ
d
θ
\int_{0}^{\pi} \frac{\sin \theta+\sin \theta \tan ^{2} \theta}{\sec ^{2} \theta} d \theta
∫
0
π
s
e
c
2
θ
s
i
n
θ
+
s
i
n
θ
t
a
n
2
θ
d
θ
Get tutor help
∫
0
1
(
a
+
(
b
−
a
)
t
)
k
d
t
,
k
∈
N
\int_{0}^{1}(a+(b-a) t)^{k} d t, k \in \mathbb{N}
∫
0
1
(
a
+
(
b
−
a
)
t
)
k
d
t
,
k
∈
N
.
Get tutor help
∫
1
/
e
tan
x
t
d
t
1
+
t
2
+
∫
1
/
e
cot
x
d
t
t
(
1
+
t
2
)
\int_{1/e}^{\tan x}\frac{t\,dt}{1+t^{2}}+\int_{1/e}^{\cot x}\frac{dt}{t(1+t^{2})}
∫
1/
e
t
a
n
x
1
+
t
2
t
d
t
+
∫
1/
e
c
o
t
x
t
(
1
+
t
2
)
d
t
Get tutor help
f
(
x
+
i
y
)
=
∑
k
=
1
∞
(
1
k
(
x
+
i
y
)
)
f(x+iy)=\sum_{k=1}^{\infty}\left(\frac{1}{k^{(x+iy)}}\right)
f
(
x
+
i
y
)
=
∑
k
=
1
∞
(
k
(
x
+
i
y
)
1
)
Get tutor help
Let
\newline
f
(
x
)
=
lim
n
→
∞
(
n
n
(
x
+
n
)
(
x
+
n
2
)
⋯
(
x
+
n
n
)
n
!
(
x
2
+
n
2
)
(
x
2
+
n
2
4
)
⋯
(
x
2
+
n
2
n
2
)
)
x
n
,
f(x)=\lim_{n \to \infty}\left(\frac{n^{n}(x+n)(x+\frac{n}{2})\cdots(x+\frac{n}{n})}{n!(x^{2}+n^{2})(x^{2}+\frac{n^{2}}{4})\cdots(x^{2}+\frac{n^{2}}{n^{2}})}\right)^{\frac{x}{n}},
f
(
x
)
=
n
→
∞
lim
(
n
!
(
x
2
+
n
2
)
(
x
2
+
4
n
2
)
⋯
(
x
2
+
n
2
n
2
)
n
n
(
x
+
n
)
(
x
+
2
n
)
⋯
(
x
+
n
n
)
)
n
x
,
for all
\newline
x
>
0.
x > 0.
x
>
0.
Then
\newline
(A)
\newline
f
(
1
2
)
≥
f
(
1
)
f\left(\frac{1}{2}\right) \geq f(1)
f
(
2
1
)
≥
f
(
1
)
\newline
(B)
\newline
f
(
1
3
)
≤
f
(
2
3
)
f\left(\frac{1}{3}\right) \leq f\left(\frac{2}{3}\right)
f
(
3
1
)
≤
f
(
3
2
)
\newline
(C)
\newline
f
′
(
2
)
≤
0
f'(2) \leq 0
f
′
(
2
)
≤
0
\newline
(D)
\newline
f
′
(
3
)
f
(
3
)
≥
f
′
(
2
)
f
(
2
)
\frac{f'(3)}{f(3)} \geq \frac{f'(2)}{f(2)}
f
(
3
)
f
′
(
3
)
≥
f
(
2
)
f
′
(
2
)
Get tutor help
Evaluate
Lim
n
→
∞
(
tan
[
π
−
4
4
+
(
1
+
1
n
)
a
]
)
n
(
α
∈
Q
)
\operatorname{Lim}_{\mathrm{n} \rightarrow \infty}\left(\tan \left[\frac{\pi-4}{4}+\left(1+\frac{1}{\mathrm{n}}\right)^{a}\right]\right)^{\mathrm{n}}(\alpha \in \mathrm{Q})
Lim
n
→
∞
(
tan
[
4
π
−
4
+
(
1
+
n
1
)
a
]
)
n
(
α
∈
Q
)
Get tutor help
Integrate.
\newline
∫
10
13
2
x
d
x
\int_{10}^{13} 2 x d x
∫
10
13
2
x
d
x
Get tutor help
1
2
3
...
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