Mathematics

# Evaluate:$\displaystyle\int{{(\ln x)}^{4}dx}$

##### SOLUTION
$\displaystyle \int (ln \, x)dx$
Let $ln\, x = t$
$x = e^t$
$dx =e^t dt$

$\displaystyle \int t^4e^tdt$

Using integration by parts,
$\displaystyle \int u.v \ dx=u\int v\ dx$ $\displaystyle -\int \left ( \dfrac{du}{dx}\int v\ dx \right )dx$

$\displaystyle = t^4 \int e^tdt - \int 4t^3e^tdt$

$\displaystyle = t^4 e^t - 4 \left[t^3 \int e^t dt - \int 3t^2 e^t dt\right]$ ..... again by parts

$\displaystyle = t^4 e^t - 4\left[t^3e^t - 3 \int t^2 e^t dt\right]$

$\displaystyle = t^4e^t -4 t^3e^t + 12\left[t^2\int e^t dt - \int 2t e^t dt\right]$ .... again by parts

$\displaystyle = t^4 e^t - 4t^3e^t+ 12 \left[ t^2e^t - 2 \int te^tdt]\right]$

$\displaystyle =t^4e^t - 4t63e^t + 12 \left[t^2e^t - 2 (te^t - \int e^tdt)\right]$ .... again by parts

$\displaystyle= t^4e^t - 4t^3e^t + 12t^2 e^t - 24te^t + 24e^t + C$

Put the value of $t$

$= (ln\, x)^4 x- 4 (ln\, x)^3 x+ 12 (ln \,x)^2 x - 24(ln\, x)x + 24x + C$

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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
Resolve into partial fraction $\displaystyle \frac{2x^2+5x-11}{x^2+2x-3}$
• A. $\displaystyle 2+\frac{2}{x+3}+\frac{1}{x-1}$
• B. $\displaystyle 2-\frac{2}{x+3}-\frac{1}{x-1}$
• C. $\displaystyle 2+\frac{1}{x+3}+\frac{1}{x-1}$
• D. $\displaystyle 2+\frac{1}{x+3}-\frac{1}{x-1}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Evaluate: $\int_{0}^{\pi}\dfrac {4x \sin x}{1 + \cos^{2}x} dx.$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\displaystyle \int \dfrac {4x+5}{2x^2+5x+18}dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Evaluate $\displaystyle \int [ \sqrt { \cot x } + \sqrt { \tan x } ] d x$

If $I= 3\displaystyle \int_{0}^{\pi }\sin ^{3}\theta \left ( 1+2\cos \theta \right )\left ( 1+\cos \theta \right )^{2}d\theta$
then the value of $I$ is?