Evaluate the integral: int frac{(x+1)(x+log x | vedclass

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

Question

(A) Let $I = \int \frac{(x+1)(x+\log x)^2}{x} dx$. Rewrite the integrand as: $I = \int \left( \frac{x}{x} + \frac{1}{x} \right) (x + \log x)^2 dx = \i

Vedclass · Accepted Answer

$\frac{(x+\log x)^3}{3} + c$

Vedclass · Answer

(A) Let $I = \int \frac{(x+1)(x+\log x)^2}{x} dx$. Rewrite the integrand as: $I = \int \left( \frac{x}{x} + \frac{1}{x} \right) (x + \log x)^2 dx = \int (1 + \frac{1}{x}) (x + \log x)^2 dx$. Let $u = x + \log x$. Then, $du = (1 + \frac{1}{x}) dx$. Substituting these into the integral: $I = \int u^2 du$. Integrating with respect to $u$: $I = \frac{u^3}{3} + c$. Substituting back $u = x + \log x$: $I = \frac{(x + \log x)^3}{3} + c$.

Evaluate the integral: $\int \frac{(x+1)(x+\log x)^2}{x} dx$

Similar Questions

$\int \frac{x^2 \tan^{-1}(x^3)}{1 + x^6} \, dx$ is equal to

$\int \frac{\left(x+\sqrt{1+x^2}\right)^2}{\sqrt{1+x^2}} d x=$

Integrate the function: $\frac{x^{2}}{\sqrt{x^{6}+a^{6}}}$

Integrate the function $\frac{\sqrt{\tan x}}{\sin x \cos x}$.

If $f(x)$ is an anti-derivative of $g(x)$ and $\int f(x) g(x) (1 + f^2(x)) dx = F(x)$,then $F(x) =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module