If int x^{x}(1+log x) d x=k x^{x}+c,then k= | vedclass

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

Question

(A) Let $I = \int x^{x}(1+\log x) dx$. Consider the substitution $u = x^{x}$. Taking the natural logarithm on both sides,$\log u = x \log x$. Differen

Vedclass · Accepted Answer

$\log _{e} e$

Vedclass · Answer

(A) Let $I = \int x^{x}(1+\log x) dx$. Consider the substitution $u = x^{x}$. Taking the natural logarithm on both sides,$\log u = x \log x$. Differentiating with respect to $x$,$\frac{1}{u} \frac{du}{dx} = 1 \cdot \log x + x \cdot \frac{1}{x} = \log x + 1$. Thus,$\frac{du}{dx} = u(1 + \log x) = x^{x}(1 + \log x)$. Therefore,$du = x^{x}(1 + \log x) dx$. Substituting this into the integral,$I = \int du = u + c$. Replacing $u$ with $x^{x}$,we get $I = x^{x} + c$. Comparing this with the given expression $k x^{x} + c$,we find $k = 1$. Since $\log_{e} e = 1$,the correct option is $A$.

If $\int x^{x}(1+\log x) d x=k x^{x}+c$,then $k=$

Similar Questions

$\int {{{\tan }^{ - 1}}\sqrt {\frac{{1 - \cos 2x}}{{1 + \cos 2x}}} } \;dx = $

$\int (\tan^7 x + \tan x) dx =$

Find the following integral:
$\int \frac{1-\sin x}{\cos ^{2} x} d x$

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

If $\int(1-\cos x) \operatorname{cosec}^2 x \, dx = f(x) + c$,then $f(x)$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module