Mathematics

# Evaluate the following integral:$\displaystyle\int{{e}^{a\log_{e}{x}}dx},\,\,a>0,a\neq\,1$

##### SOLUTION
$\displaystyle\int{{e}^{a\log_{e}{x}}dx},\,\,a>0,a\neq\,1$

$=\displaystyle\int{{e}^{\log_{e}{{x}^{a}}}dx}$ since ${a}^{\log_{a}{N}}=N$

$=\displaystyle\int{{x}^{a}\,dx}$

$=\dfrac{1}{a+1}{x}^{a+1}+c$ since $\displaystyle\int{{x}^{n}\,dx}=\dfrac{{x}^{n+1}}{n+1}+c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
If $\smallint \cfrac{{{e^x} - 1}}{{{e^x} + 1}}dx = f\left( x \right) + C$, then f(x) is equal to
• A. $2\log \left| {{e^x} + 1} \right|$
• B. $\log \left| {{e^{2x}} - 1} \right|$
• C. $\log \left| {{e^{2x}} + 1} \right|$
• D. $e^{x}-2\log \left| {{e^x} + 1} \right|$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate :
$\int \dfrac {2 \cos x + 3 \sin x }{4 \cos x + 5 \sin x} dx.$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\displaystyle\int\limits_{0}^{2\pi}2\sin x \ dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate $\displaystyle \int { \frac { 1 }{(1-2x) } }dx$

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 $\displaystyle f(x)>0 \forall x \in R$