Find int(1)/(5x^(2)-20 x+100)dx. Choose 1 answer: (A) arctan((x-2)/(4))+C (B) arcsin((x-2)/(4))+C (c) (1)/(20)arcsin((x-2)/(4))+C (D) (1)/(20)arctan((x-2)/(4))+C

Rewrite Quadratic Denominator: Rewrite the quadratic in the denominator to complete the square: 5(x^2 - 4x + 20).Complete Square: Complete the square for the expression x^2 - 4x + 20 by adding and subtracting (4/2)^2 = 4 inside the parenthesis: 5((x^2 - 4x + 4) + 16).Factor Out Constant: Factor out the 5 and rewrite the expression as 5((x - 2)^2 + 16).Standard Form Integral: Now the integral looks like int(1)/(5((x-2)^2+16))dx, which is a standard form for the arctangent function derivative.Pull Out Constant: Pull out the constant 1/5 from the integral: (1/5) * int(1)/((x-2)^2+16)dx.Recognize Integral Form: Recognize that the integral is now in the form of int(1)/(a^2 + u^2)du, which is 1/a * arctan(u/a) + C.Substitute Constants: In our case, a = 4 and u = x - 2. So the integral becomes (1/5) * (1/4) * arctan((x-2)/4) + C.Simplify Constants: Simplify the constants: (1/20) * arctan((x-2)/4) + C.

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Find
int(1)/(5x^(2)-20 x+100)dx.
Choose 1 answer:
(A)
arctan((x-2)/(4))+C
(B)
arcsin((x-2)/(4))+C
(c)
(1)/(20)arcsin((x-2)/(4))+C
(D)
(1)/(20)arctan((x-2)/(4))+C

Find $\int \frac{1}{5 x^{2}-20 x+100} d x$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\arctan \left(\frac{x-2}{4}\right)+C$ $\newline$ (B) $\arcsin \left(\frac{x-2}{4}\right)+C$ $\newline$ (c) $\frac{1}{20} \arcsin \left(\frac{x-2}{4}\right)+C$ $\newline$ (D) $\frac{1}{20} \arctan \left(\frac{x-2}{4}\right)+C$

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Find derivatives of using multiple formulae

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Q. Find

\int \frac{1}{5 x^{2}-20 x+100} d x

\newline

Choose

1

answer:

\newline

(A)

\arctan \left(\frac{x-2}{4}\right)+C

\newline

(B)

\arcsin \left(\frac{x-2}{4}\right)+C

\newline

(c)

\frac{1}{20} \arcsin \left(\frac{x-2}{4}\right)+C

\newline

(D)

\frac{1}{20} \arctan \left(\frac{x-2}{4}\right)+C

Rewrite Quadratic Denominator: Rewrite the quadratic in the denominator to complete the square: $5(x^2 - 4x + 20)$ .
Complete Square: Complete the square for the expression $x^2 - 4x + 20$ by adding and subtracting $(\frac{4}{2})^2 = 4$ inside the parenthesis: $5((x^2 - 4x + 4) + 16)$ .
Factor Out Constant: Factor out the $5$ and rewrite the expression as $5((x - 2)^2 + 16)$ .
Standard Form Integral: Now the integral looks like $\int \frac{1}{5((x-2)^2+16)}dx$ , which is a standard form for the arctangent function derivative.
Pull Out Constant: Pull out the constant $\frac{1}{5}$ from the integral: $\left(\frac{1}{5}\right) \cdot \int \frac{1}{(x-2)^2+16}\,dx$ .
Recognize Integral Form: Recognize that the integral is now in the form of $\int \frac{1}{a^2 + u^2} \, du$ , which is $\frac{1}{a} \cdot \arctan(\frac{u}{a}) + C$ .
Substitute Constants: In our case, $a = 4$ and $u = x - 2$ . So the integral becomes $(1/5) \times (1/4) \times \arctan((x-2)/4) + C$ .
Simplify Constants: Simplify the constants: $(\frac{1}{20}) \cdot \arctan(\frac{x-2}{4}) + C$ .