int frac{x^{9/2}}{sqrt{1+x^{11}}} dx is equal | vedclass

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

Question

(A) Let $I = \int \frac{x^{9/2}}{\sqrt{1+x^{11}}} dx$. Substitute $t = x^{11/2}$. Then,$dt = \frac{11}{2} x^{9/2} dx$,which implies $x^{9/2} dx = \fra

Vedclass · Accepted Answer

$\frac{2}{11} \log \left(x^{11/2}+\sqrt{1+x^{11}}\right)+c$

Vedclass · Answer

(A) Let $I = \int \frac{x^{9/2}}{\sqrt{1+x^{11}}} dx$. Substitute $t = x^{11/2}$. Then,$dt = \frac{11}{2} x^{9/2} dx$,which implies $x^{9/2} dx = \frac{2}{11} dt$. Substituting these into the integral: $I = \int \frac{\frac{2}{11} dt}{\sqrt{1+t^2}} = \frac{2}{11} \int \frac{dt}{\sqrt{1+t^2}}$. Using the standard integral formula $\int \frac{dx}{\sqrt{1+x^2}} = \log(x + \sqrt{1+x^2}) + c$: $I = \frac{2}{11} \log(t + \sqrt{1+t^2}) + c$. Substituting back $t = x^{11/2}$: $I = \frac{2}{11} \log(x^{11/2} + \sqrt{1+x^{11}}) + c$.

$\int \frac{x^{9/2}}{\sqrt{1+x^{11}}} dx$ is equal to -

Similar Questions

If $\int \frac{\log \left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}} dt=\frac{1}{2}(g(t))^2+c$ where $c$ is a constant of integration,then $g(2)$ is equal to

The value of $\int {\left( {1 + \frac{1}{{{x^2}}}} \right){e^{\left( {x - \frac{1}{x}} \right)}}} \,dx$ equals

$\int \frac{dx}{e^x + e^{-x}} = $

If $f(x) = \int \frac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} dx, x \geq 0$ and $f(0) = 0$,then the value of $f(1)$ is

For ${x^2} \ne n\pi + 1, n \in N$ (the set of natural numbers),the integral $\int {x\sqrt {\frac{{2\sin \left( {{x^2} - 1} \right) - \sin 2\left( {{x^2} - 1} \right)}}{{2\sin \left( {{x^2} - 1} \right) + \sin 2\left( {{x^2} - 1} \right)}}} } dx$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module