int {{x^x}(1 + log x),dx} is equal to | 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 function $f(x) = x^x$. Taking the natural logarithm on both sides,we get $\log f(x) = x \log

Vedclass · Accepted Answer

${x^x} + C$

Vedclass · Answer

(A) Let $I = \int {{x^x}(1 + \log x)\,dx} $. Consider the function $f(x) = x^x$. Taking the natural logarithm on both sides,we get $\log f(x) = x \log x$. Differentiating both sides with respect to $x$,we get $\frac{1}{f(x)} f'(x) = 1 \cdot \log x + x \cdot \frac{1}{x} = \log x + 1$. Thus,$f'(x) = f(x)(1 + \log x) = x^x(1 + \log x)$. Since the derivative of $x^x$ is $x^x(1 + \log x)$,the integral of $x^x(1 + \log x)$ is $x^x + C$.

$\int {{x^x}(1 + \log x)\,dx} $ is equal to

Similar Questions

Find the anti-derivative $F$ of $f$ defined by $f(x) = 4x^{3} - 6$,where $F(0) = 3$.

The integral $\int \sqrt{1 + 2\cot x(\csc x + \cot x)} \,dx$ for $0 < x < \frac{\pi}{2}$ is equal to (where $C$ is a constant of integration):

$\int \frac{d x}{\cos (x+4) \cos (x+2)}$ is equal to

$\int \frac{x^5}{x^2+1} \, dx =$

The value of $ \int \frac{e^{6 \log x}-e^{5 \log x}}{e^{4 \log x}-e^{3 \log x}} dx $ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module