Mathematics

# Integrate the following:$\displaystyle \int\limits_0^{\pi /2} {{{\sin }^3}x} \,\,\cos x\,\,dx$

##### SOLUTION
$I = \displaystyle \int^{\pi/2}_0\sin^3c\cos x\ dx$

$= \displaystyle \int_0^{\pi/2} \sin^3\left(\dfrac{\pi}{2}-x\right) \cos \left(\dfrac{\pi}{2} - x\right)dx$

$= \displaystyle \int_{0}^{\pi/2} \cos^3x\sin x\, dx$

$\therefore I + I = \displaystyle \int_0^{\pi/2} (\sin^3x\cos x + \cos^3x\sin x)dx$

$2I = \int_0^{\pi/2}\sin x \cos x(\sin^2x + \cos^2x)dx$

$= \int_0^{\pi/2} \sin x \cos x dx$

$= \left[ \dfrac{\sin^2x}{2}\right]_0^{\pi/2} = \dfrac{1}{2}$

$I = \dfrac{1}{4}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int\frac{(x-1)}{(x+1)(x^{2}+1)}dx=$
• A. $\displaystyle \frac{1}{2}\log|\displaystyle \frac{x+1}{\sqrt{x^{2}+1}}|+c$
• B. $\displaystyle \log|\frac{\sqrt{x^{2}+1}}{x+1}|+c$
• C. $\log|x+1 |+\displaystyle \frac{1}{2}\tan^{-1}{x}+c$
• D. $-\log|x+1 |+\tan ^{-1}{x} + c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Find $\displaystyle \int \frac{dx}{x\left ( x^{3}+1 \right )^{2}}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\int \dfrac {1}{\sqrt {3x+5}-\sqrt {3x+2}}dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
. Let $\displaystyle \mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{(1+\mathrm{x}^{\mathrm{n}})^{1/\mathrm{n}}}$ for $\mathrm{n}\geq 2$ and $\displaystyle \mathrm{g}(\mathrm{x})=\frac{(\mathrm{f}\mathrm{o}\mathrm{f}\mathrm{o}\ldots \mathrm{o}\mathrm{f})}{\mathrm{f}\mathrm{o}\mathrm{c}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{s}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}}(\mathrm{x})$ . Then $\displaystyle \int \mathrm{x}^{\mathrm{n}-2}\mathrm{g}$ (x)dx equals

• A. $\displaystyle \frac{1}{\mathrm{n}-1}(1+\mathrm{n}\mathrm{x}^{\mathrm{n}})^{1\frac{1}{\mathrm{n}}}+\mathrm{K}$
• B. $\displaystyle \frac{1}{\mathrm{n}(\mathrm{n}+1)}(1+\mathrm{n}\mathrm{x}^{\mathrm{n}})^{1+\frac{1}{\mathrm{n}}}+\mathrm{K}$
• C. $\displaystyle \frac{1}{\mathrm{n}+1}(1+\mathrm{n}\mathrm{x}^{\mathrm{n}})^{1+\frac{1}{\mathrm{n}}}+\mathrm{K}$
• D. $\displaystyle \frac{1}{\mathrm{n}(\mathrm{n}-1)}(1+\mathrm{n}\mathrm{x}^{\mathrm{n}})^{1\frac{1}{\mathrm{n}}}+\mathrm{K}$

$\displaystyle \int \dfrac{8x+5}{4x^2+5x+6} dx$