Mathematics

# Evaluate:$\int { \sin ^{ 4 }{ x } } \cos ^{ 3 }{ x } .dx$

##### SOLUTION
$2sinccosD=sin\left\{ C-D \right\} +sin\left\{ C+D \right\}$
$\int { sin4xcos3xdx=\int { \frac { 1 }{ 2 } \left[ sin\left\{ 4x-3x \right\} +sin\left\{ 4x+3x \right\} \right] } dx }$
$\frac { 1 }{ 2 } \int { \left[ sinx+sin7x \right] dx }$
$\frac { 1 }{ 2 } \left[ -cosx-\frac { cos7x }{ 7 } \right] C$
$-\frac { 1 }{ 2 } cosx-\frac { 1 }{ 14 } cos7x+C$

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Subjective Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

#### Realted Questions

Q1 Subjective Hard
Evaluate
$\displaystyle \int \dfrac{\sin x}{\sin (x-a)}dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard

lf $f(x)=p(x)q(x)$ where $p(x)=\sqrt{\cos x},\ q(x)=\displaystyle \log(\frac{1-x}{1+x})$ then $\displaystyle \int^{b}_{a} f(x)dx$ equals where $a=-1/2 b=1/2$
• A. $2\displaystyle \int_{0}^{1/2}p(x)q(x)dx$
• B. $\displaystyle \int_{0}^{b}p(x)q(x)$
• C. 1
• D.

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium

$\displaystyle \int_{-a}^{a}\frac{x^{4}dx}{\sqrt{a^{2}-x^{2}}}=$
• A. $\displaystyle \frac{\pi a^{4}}{8}$
• B. $\displaystyle \frac{-\pi a^{4}}{8}$
• C. $\displaystyle \frac{5\pi \mathrm{a}^{4}}{8}$
• D. $\displaystyle \frac{3\pi a^{4}}{8}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
Evaluate $\displaystyle \int_{0}^{2\pi }e^{x}\cos \left ( \frac{\pi }{4}+\frac{x}{2} \right )dx$
• A. $\displaystyle -\frac{3\sqrt{2}}{5}\left ( e^{2\pi }-1 \right )$
• B. $\displaystyle \frac{3\sqrt{2}}{5}\left ( e^{2\pi }-1 \right )$
• C. $\displaystyle \frac{3\sqrt{2}}{5}\left ( e^{2\pi }+1 \right )$
• D. $\displaystyle -\frac{3\sqrt{2}}{5}\left ( e^{2\pi }+1 \right )$

Prove that $\displaystyle\int_0^{\pi/2}$ $ln(\sin x)dx=\displaystyle\int_0^{\pi/2}ln(cos x)dx=\int_0^{\pi/2}\,\,ln(sin2x)dx=-\dfrac{\pi}{2}.ln 2$.