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Find
int(1)/(sqrt(-x^(2)+10 x+11))dx.
Choose 1 answer:
(A)
(1)/(6)arctan((x-5)/(6))+C
(B)
(1)/(6)arcsin((x-5)/(6))+C
(C)
arcsin((x-5)/(6))+C
(D)
arctan((x-5)/(6))+C

Find $\int \frac{1}{\sqrt{-x^{2}+10 x+11}} d x$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{6} \arctan \left(\frac{x-5}{6}\right)+C$ $\newline$ (B) $\frac{1}{6} \arcsin \left(\frac{x-5}{6}\right)+C$ $\newline$ (C) $\arcsin \left(\frac{x-5}{6}\right)+C$ $\newline$ (D) $\arctan \left(\frac{x-5}{6}\right)+C$

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Evaluate definite integrals using the chain rule

Full solution

Q. Find

\int \frac{1}{\sqrt{-x^{2}+10 x+11}} d x

.

\newline

Choose

1

answer:

\newline

(A)

\frac{1}{6} \arctan \left(\frac{x-5}{6}\right)+C

\newline

(B)

\frac{1}{6} \arcsin \left(\frac{x-5}{6}\right)+C

\newline

(C)

\arcsin \left(\frac{x-5}{6}\right)+C

\newline

(D)

\arctan \left(\frac{x-5}{6}\right)+C

Recognize Integral Form: Next, we recognize that the integral is in the form of an arcsin function, where the integral of $\frac{1}{\sqrt{a^2 - u^2}} \, du$ is $\arcsin(\frac{u}{a}) + C$ . Here, $a = 6$ and $u = x - 5$ . So, the integral becomes: $(\frac{1}{6})\arcsin(\frac{x - 5}{6}) + C$
Substitute Values: Now we check the answer choices to see which one matches our result. $\newline$ The correct answer is (B) $(\frac{1}{6})\arcsin(\frac{(x - 5)}{6}) + C$ .

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