Mathematics

# Solve :$\int x^x ln (ex) dx$

##### SOLUTION
$\begin{array}{l} \int { { x^{ x } }\left( { \ln { \left( { ex } \right) } } \right) } dx \\ \int { { x^{ x } }\left( { 1+\ln { x } } \right) } dx \\ Let\, { x^{ x } }=t \\ \log t=x\log x \\ \frac { 1 }{ t } =1+\log x \\ \frac { { dt } }{ { dx } } ={ x^{ x } }\left( { 1+\log x } \right) \\ \int { dt } =t+c={ x^{ x } }+C \end{array}$
Hence,
Solved.

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
If $f(x)=\begin{vmatrix}\sin x+\sin2x+\sin3x & \sin2x & \sin3x\\ 3+4\sin x & 3 & 4\sin x\\ 1+\sin x & \sin x & 1\end{vmatrix}$, then the value of $\displaystyle \int_{0}^{\frac{\pi}2}f(x)dx$, is
• A. $3$
• B. $\dfrac23$
• C. $0$
• D. $\dfrac13$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\int \displaystyle \frac {cos^2\, x}{1+tan\, x}\, dx$
• A. $\displaystyle \frac {1}{4}\, ln\, (cos\, -\, sin\, x)\, +\, \displaystyle \frac {x}{2}\, +\, \displaystyle \frac {1}{8}\, (sin\, 2x\, -\, cos\, 2x)$
• B. $\displaystyle \frac {1}{4}\, ln\, (cos\, +\, sin\, x)\, +\, \displaystyle \frac {x}{2}\, +\, \displaystyle \frac {1}{8}\, (sin\, 2x\, -\, cos\, 2x)$
• C. $\displaystyle \frac {1}{4}\, ln\, (cos\, -\, sin\, x)\, +\, \displaystyle \frac {x}{2}\, +\, \displaystyle \frac {1}{8}\, (sin\, 2x\, +\, cos\, 2x)$
• D. $\displaystyle \frac {1}{4}\, ln\, (cos\, +\, sin\, x)\, +\, \displaystyle \frac {x}{2}\, +\, \displaystyle \frac {1}{8}\, (sin\, 2x\, +\, cos\, 2x)$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Prove that $\displaystyle \int \dfrac {f'(x)}{\sqrt [n]{f(x)}} dx = \dfrac {[f(x)]^{1 - \tfrac {1}{n}}}{1 - \dfrac {1}{n}} + C, n\neq 1$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following integral:
$\displaystyle\int^1_0x^3\sqrt{1+3x^4}dx$.

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Medium
Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$

On the basis of above information, answer the following questions :

Asked in: Mathematics - Limits and Derivatives

1 Verified Answer | Published on 17th 08, 2020