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

Question

Quizplus · Accepted Answer

The integral $\int \cot^{-1} x dx$ can be evaluated using the given formula. The derivative of $\cot^{-1} x$ with respect to $x$ is $-\frac{1}{1+x^2}$. Substituting this into the formula gives $x \cot^{-1} x + \int \frac{dx}{1+x^2}$, which simplifies to $x \cot^{-1} x + \ln \sqrt{x^2 + 1} + C$.

Evaluate the Integral ∫f−1(x)dx=xf−1(x)−∫x(ddxf−1(x))dx\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int x \left( \frac { d } { d x } f ^ { - 1 } ( x ) \right) d x∫f−1(x)dx=xf−1(x)−∫x(dxd​f−1(x))dx

Evaluate the Integral $\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int x \left( \frac { d } { d x } f ^ { - 1 } ( x ) \right) d x$