Mathematics

# $\displaystyle \int { \dfrac { { e }^{ 5\log { x } }-{ e }^{ 4\log { x } } }{ { e }^{ 3\log { x } }-{ e }^{ 2\log { x } } } } dx=\_ \_ \_ \_ \_ \_ +c$

$\dfrac{{ x }^{ 3 }} 3$

##### SOLUTION
Given,

$\displaystyle \int \dfrac{e^{5\ln \left(x\right)}-e^{4\ln \left(x\right)}}{e^{3\ln \left(x\right)}-e^{2\ln \left(x\right)}}dx$

Now,

$\displaystyle \dfrac{e^{5\ln \left(x\right)}-e^{4\ln \left(x\right)}}{e^{3\ln \left(x\right)}-e^{2\ln \left(x\right)}}$

$\displaystyle=\dfrac{e^{4\ln \left(x\right)}\left(e^{\ln \left(x\right)}-1\right)}{e^{3\ln \left(x\right)}-e^{2\ln \left(x\right)}}$

$\displaystyle=\dfrac{e^{4\ln \left(x\right)}\left(e^{\ln \left(x\right)}-1\right)}{e^{2\ln \left(x\right)}\left(e^{\ln \left(x\right)}-1\right)}$

$\displaystyle=\dfrac{e^{4\ln \left(x\right)}}{e^{2\ln \left(x\right)}}$

$\displaystyle=\dfrac{e^{4\ln \left(x\right)}}{x^2}$

$\displaystyle=\dfrac{x^4}{x^2}=x^2$

$\displaystyle \therefore \int \dfrac{e^{5\ln \left(x\right)}-e^{4\ln \left(x\right)}}{e^{3\ln \left(x\right)}-e^{2\ln \left(x\right)}}dx=\int x^2 dx$

$=\dfrac{x^{2+1}}{2+1}$

$=\dfrac{x^3}{3}+C$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 114

#### Realted Questions

Q1 Single Correct Medium
For any integer n the integral $\displaystyle \int_{0}^{\pi }e^{\cos ^{2}x}\cos ^{3}\left ( 2n+1 \right )xdx$ has the value
• A. $\displaystyle \pi$
• B. $1$
• C. none of these
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
The minimum value of the function f(x) = $\int^x_0 \frac{d \theta}{cos \theta} + \int^{\pi/2}_x \frac{d \theta}{sin \theta}$ where $x \in [0, \frac{\pi}{2}],$ is
• A. $ln(2\sqrt{2} + 2)$
• B. $ln(\sqrt{3} + 2)$
• C. $ln(\sqrt{2} + 3)$
• D. $2ln(\sqrt{2} + 1)$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Solve $\displaystyle\int \frac{\sec ^{4}x}{\sqrt{\left ( \tan x \right )}}dx$
• A. $\displaystyle \frac{2}{5}\sqrt{\tan x}\left ( 5-\tan ^{2}x\right ).$
• B. $\displaystyle \frac{2}{7}\sqrt{\tan x}\left ( 5+\tan ^{2}x\right ).$
• C. $\displaystyle \frac{2}{7}\sqrt{\tan x}\left ( 5+\tan ^{3}x\right ).$
• D. $\displaystyle \frac{2}{5}\sqrt{\tan x}\left ( 5+\tan ^{2}x\right ).$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Find the value of $\displaystyle\int\limits_{-\dfrac{\pi}{2}}^{\dfrac{\pi}{2}}|\sin x|\ dx$.

If $I=\displaystyle \int_{0}^{\pi/2}sinx.log(sin x)dx = log\left(\dfrac{K}{e}\right).$ Then find the value of $K.$