Evaluate the integral.
-Use the formula $\int \mathrm { f } ^ { - 1 } ( \mathrm { x } ) \mathrm { dx } = \mathrm { xf } ^ { - 1 } ( \mathrm { x } ) - \int \mathrm { x } \left( \frac { \mathrm { d } } { \mathrm { dx } } \mathrm { f } ^ { - 1 } ( \mathrm { x } ) \right) \mathrm { dx }$ to evaluate the integral. $\int \cot ^ { - 1 } x d x$
A) $x \cot ^ { - 1 } x + \ln \sqrt { x ^ { 2 } + 1 } + C$
B) $x \cot ^ { - 1 } x + \ln x + C$
C) $x \cot ^ { - 1 } x + x + C$
D) $x \cot ^ { - 1 } x - \ln \sqrt { x ^ { 2 } + 1 } + C$

Question

Quizplus · Accepted Answer

The integral $\int \cot ^ { - 1 } x d x$ can be evaluated using the given formula, where $f^{-1}(x) = \cot^{-1}x$. The derivative of $\cot^{-1}x$ with respect to $x$ is $-\frac{1}{1+x^2}$. Substituting these into the formula gives $x \cot^{-1}x + \ln\sqrt{x^2 + 1} + C$, which matches choice A.

Evaluate the Integral To Evaluate the Integral ∫cot⁡−1xdx\int \cot ^ { - 1 } x d x∫cot−1xdx

Evaluate the Integral To Evaluate the Integral $\int \cot ^ { - 1 } x d x$