If int f(x) , dx = g(x),then int f^{-1}(x) , | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given that $\int f(x) \, dx = g(x)$.We need to evaluate $\int f^{-1}(x) \, dx$.Using integration by parts,let $u = f^{-1}(x)$ and $dv = dx$. Then

Vedclass · Accepted Answer

$x f^{-1}(x) - g(f^{-1}(x))$

Vedclass · Answer

(B) Given that $\int f(x) \, dx = g(x)$.We need to evaluate $\int f^{-1}(x) \, dx$.Using integration by parts,let $u = f^{-1}(x)$ and $dv = dx$. Then $du = \frac{d}{dx} f^{-1}(x) \, dx$ and $v = x$.$\int f^{-1}(x) \, dx = x f^{-1}(x) - \int x \frac{d}{dx} f^{-1}(x) \, dx$.Let $f^{-1}(x) = t$,then $x = f(t)$ and $dx = f'(t) \, dt$.Also,$\frac{d}{dx} f^{-1}(x) \, dx = dt$.Substituting these into the integral:$\int f^{-1}(x) \, dx = x f^{-1}(x) - \int f(t) \, dt$.Since $\int f(t) \, dt = g(t)$,we have:$\int f^{-1}(x) \, dx = x f^{-1}(x) - g(t) = x f^{-1}(x) - g(f^{-1}(x))$.Thus,the correct option is $B$.

If $\int f(x) \, dx = g(x)$,then $\int f^{-1}(x) \, dx$ is equal to

Similar Questions

If $\int {e^x \sin x \, dx} = \frac{1}{2} e^x \cdot a + c$,then $a = $

Integrate the function: $e^{2x} \sin x$

$\int \frac{x^3}{\sqrt{1+x^2}} dx$ is equal to

If $\int f(x) dx = g(x)$, then $\int x^3 f(x^2) dx$ is equal to

$\int x \sec^2 x \, dx = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module