Mathematics

# Evaluate the given integral.$\int { { x }^{ 3 }\log { x } } dx$

##### SOLUTION
$I=\displaystyle\int{{x}^{3}\log{x}dx}$

Integrating by parts, we get

Let $dv={x}^{3}dx\Rightarrow\,v=\dfrac{{x}^{4}}{4}$

$u=\log{x}\Rightarrow\,du=\dfrac{1}{x}dx$

$\int u.v dx=u \int vdx-\int \left [\int vdx. \dfrac{du}{dx}.dx \right ]$......by parts formula.

$\Rightarrow\,I=\dfrac{{x}^{4}\log{x}}{4}-\displaystyle\int{\dfrac{{x}^{4}}{4}\times\dfrac{1}{x}dx}$

$\Rightarrow\,I=\dfrac{{x}^{4}\log{x}}{4}-\dfrac{1}{4}\displaystyle\int{{x}^{3}dx}$

$\Rightarrow\,I=\dfrac{{x}^{4}\log{x}}{4}-\dfrac{1}{4}\dfrac{{x}^{4}}{4}+c$

$\therefore\,I=\dfrac{{x}^{4}\log{x}}{4}-\dfrac{{x}^{4}}{16}+c$

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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
$\displaystyle\int \frac{1}{\sqrt{\left ( x^{2}+3x+1 \right )}}dx.$
• A. $\displaystyle \log \left [ \left ( x-3/2 \right ) \right ]+\sqrt{\left \{ \left ( x-3/2 \right )^{2}-\left ( \sqrt{3/2} \right )^{2} \right \}}.$
• B. $\displaystyle \log \left [ \left ( x+3/2 \right ) \right ]-\sqrt{\left \{ \left ( x+3/2 \right )^{2}-\left ( \sqrt{5/2} \right )^{2} \right \}}.$
• C. $\displaystyle \log \left [ \left ( x-3/2 \right ) \right ]-\sqrt{\left \{ \left ( x-3/2 \right )^{2}-\left ( \sqrt{3/2} \right )^{2} \right \}}.$
• D. $\displaystyle \log \left [ \left ( x+3/2 \right ) \right ]+\sqrt{\left \{ \left ( x+3/2 \right )^{2}-\left ( \sqrt{5/2} \right )^{2} \right \}}.$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate $\int \dfrac {(1+2x^{2})dx}{x^{2}(1+x^{2})}$.

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Evaluate: $\displaystyle \int { \dfrac { { x }^{ 2 } }{ 1+{ x }^{ 4 } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Integrate $\displaystyle \frac {1}{2}\, f\, '(x)\, w.r.t.\, x^4$, where $f(x)\, =\, tan^{-1}\, x\, +\, ln\, \sqrt{1\, +\, x}\, -\, ln\, \sqrt{1\, -\, x}$
• A. $\, -\, ln(1+x^2)\, +\, C$
• B. $\, -\, ln(1-x^2)\, +\, C$
• C. $\, -\, ln(1+x^4)\, +\, C$
• D. $\, -\, ln(1-x^4)\, +\, C$

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