Mathematics

# $\int _{ 0 }^{ \pi /4 }{ \sec ^{ 7 }{ \theta } \sin ^{ 3 }{ \theta } d\theta }$

$5/12$

##### SOLUTION
Let $\displaystyle I=\int_{0}^{\dfrac{\pi }{4}} \sec^{7}\theta \sin^{3}\theta d\theta$
$=\displaystyle \int_{0}^{\dfrac{\pi }{4}}\sec^{4}\theta \tan^{3}\theta d\theta$
$=\displaystyle \int_{0}^{\dfrac{\pi }{4}}\sec^{2}\theta \tan^{5}\theta +\int_{0}^{\dfrac{\pi }{4}}\tan^{3}\theta \sec^{2}\theta d\theta$
Let, $\tan\theta =4, du=\sec^{2}\theta d\theta$
$\theta \epsilon \left(0,\dfrac{\pi }{4}\right),4\epsilon (0,1)$
$\therefore \displaystyle I=\int_{0}^{1}x^{5}du+\int_{0}^{1}u^{3}du$
$I=\displaystyle \frac{1}{6}+\frac{1}{4}=\frac{5}{12}$

Its FREE, you're just one step away

Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Hard
$\displaystyle\int_{0}^{\pi}\frac{x^{2}\cos x}{\left ( 1+\sin x \right )^{2}} dx$ equals
• A. $\pi \left ( \pi -2 \right )$
• B. $\pi^{2} \left ( \pi +2 \right )$
• C. None of these
• D. $\pi \left (2- \pi \right )$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate the following integral
$\int { \cfrac { 1 }{ \sqrt { x } \left( \sqrt { x } +1 \right) } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
By Simpson's rule, the value of $\displaystyle\int _{ -3 }^{ 3 }{ { x }^{ 4 }dy }$ by taking 6 sub-intervals, is
• A. $90$
• B. $80$
• C. $70$
• D. $98$

1 Verified Answer | Published on 17th 09, 2020

Q4 One Word Medium
$\displaystyle \int_{0}^{\pi /2}\sin 2x\log \tan x dx=$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
The limit of the area under the curve $y=e^{-x}$ from $x=0$ to $x=h$ at $h\rightarrow \infty$ is
• A. $2$
• B. $e$
• C. $1/e$
• D. $1$