Mathematics

# Evaluate:$\int sin^4 x cos^4x dx$

##### SOLUTION
$I=\displaystyle \int \sin^4x. \cos^4x.dx$

We know that

$\sin^2x=\dfrac {1-\cos 2x}{2},\ \cos^2x=\dfrac {1+\cos 2x}{2}$

$\displaystyle \int \left (\dfrac {1-\cos 2x}{2}\right)^2. \left (\dfrac {1+\cos 2x}{2}\right)^2dx$

$=\dfrac {1}{16}\displaystyle \int (1-\cos^22x)^2dx$

$=\dfrac {1}{16}\displaystyle \int (1+\cos^42x-2\cos^2 2x)dx$

$=\dfrac {1}{16}\displaystyle \int 1+\dfrac {(1+\cos 4x)^2}{4}-2\left (\dfrac {1+\cos 4x}{2}\right)dx$

$=\dfrac {1}{16}\displaystyle \int 1+\dfrac {1+\cos^24x+2\cos 4x}{2}-1-\cos 4x \ dx$

$=\dfrac {1}{16}\displaystyle \int \dfrac {1+\dfrac {1+\cos 8x}{2}+2\cos 4x-2\cos 4x}{2}dx$

$=\dfrac {1}{32}\displaystyle \int 1+\dfrac {1+\cos 8x}{2}dx$

$=\dfrac {1}{64}\displaystyle \int 3+\cos 8x\ dx$

$=\dfrac {1}{64} \left [3x+\dfrac {\sin 8x}{8}\right]+c$

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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 Hard
The value of integral $\displaystyle \int \frac{\left(\sqrt{1+x^{2}}+x \right)^{n}}{\sqrt{1+x^{2}}}dx$, is
• A. $\displaystyle \frac{1}{n}\left ( \sqrt{1+x^{2}}+x \right )^{n-1}+c$
• B. $\displaystyle \frac{1}{n-1}\left ( \sqrt{1+x^{2}}+x \right )^{n-1}+c$
• C. $\displaystyle \frac{1}{n-1}\left ( \sqrt{1+x^{2}}+x \right )^{n}+c$
• D. $\displaystyle \frac{1}{n}\left ( \sqrt{1+x^{2}}+x \right )^{n}+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Integrate the rational function   $\cfrac {1}{x(x^n+1)}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
If y=${\left( {\log x} \right)^x}\frac{{dy}}{{dx}}$ then $\frac{{dy}}{{dx}}$
• A. ${\left( {\log x} \right)^x}\left[ {\log \left( {\log x} \right) - \frac{1}{{\log x}}} \right]$
• B. $- {\left( {\log x} \right)^x}\left[ {\log \left( {\log x} \right) + \frac{1}{{\log x}}} \right]$
• C. ${\left( {\log x} \right)^x}\left[ {-\log \left( {\log x} \right) + \frac{1}{{\log x}}} \right]$
• D. ${\left( {\log x} \right)^x}\left[ {\log \left( {\log x} \right) + \frac{1}{{\log x}}} \right]$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\int { P\left( x \right) { e }^{ kx }dx=Q\left( x \right) { e }^{ 4x }+C }$, where $P(x)$ is polynomial of degree $n$ and $Q(x)$ is polynomial of degree $7$. Then the value of $n+7+k+\lim _{ x\rightarrow \infty }{ \dfrac { P\left( x \right) }{ Q\left( x \right) } }$ is:
• A. $18$
• B. $19$
• C. $20$
• D. $22$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\int e^{2ax}\dfrac {1-\cos 2a x}{1+\sin 2ax}dx$ is equal to
• A. $-\dfrac {1}{a}e^{2ax}\cos \left(\dfrac {\pi}{4}+ax\right)+C$
• B. $-\dfrac {1}{2a}e^{2ax}\cos \left(\dfrac {\pi}{4}+ax\right)+C$
• C. $-\dfrac {1}{a}e^{2ax}\csc \left(\dfrac {\pi}{4}+ax\right)+C$
• D. $-\dfrac {1}{a}e^{2ax}\cot \left(\dfrac {\pi}{4}+ax\right)+C$