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

The value of $\lambda$ for which $\int \frac{4x^3 + \lambda 4^x}{4^x + x^4} \, dx = \log (4^x + x^4) + c$ is

$\int \frac{e^x-1}{e^x+1} dx =$

If $f\left(\frac{2 x+3}{3 x+5}\right)=x+4$, where $x \neq \frac{-5}{3}, \frac{-2}{3}$, and $\int f(x) d x=A x+B \ln |3 x-2|+C$, then $3 B-A=$

$\int \frac{dx}{a^2 - x^2}$ is equal to

If $\int \frac{\cos 3x}{\sin x} dx = p \cos 2x + q \log |\sin x| + C$,then $p + q =$ . . . . . . .

Vedclass Test Series

Exam Paper Generator

Online Exam Module