Evaluate the integral.
-Use the formula $\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int f ( y ) d y , y = f ^ { - 1 } ( x )$ to evaluate the integral.
$\int \cot ^ { - 1 } x d x$
A) $x \cot ^ { - 1 } x - \ln \sin \left( \cot ^ { - 1 } x \right) + C$
B) $x \cot ^ { - 1 } x - \ln \sin x + C$
C) $x \cot ^ { - 1 } x - x + C$
D) $x \cot ^ { - 1 } x + \ln \sin \left( \cot ^ { - 1 } x \right) + C$

Question

Quizplus · Accepted Answer

Using the given formula, we have $f^{-1}(x) = \cot^{-1}x$, so $f(y) = \cot y$. The integral of $\cot y$ is $\ln|\sin y|$. Substituting $y = \cot^{-1}x$ gives the integral as $x \cot^{-1}x - \ln|\sin(\cot^{-1}x)| + C$, which matches choice A.

Evaluate the Integral ∫f−1(x)dx=xf−1(x)−∫f(y)dy,y=f−1(x)\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int f ( y ) d y , y = f ^ { - 1 } ( x )∫f−1(x)dx=xf−1(x)−∫f(y)dy,y=f−1(x)

Evaluate the Integral $\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int f ( y ) d y , y = f ^ { - 1 } ( x )$