Mathematics

# Evaluate the following integrals:$\displaystyle \int { \sec ^{ 6 }{ x } \tan ^{ }{ x } } dx$

##### SOLUTION
$\displaystyle\int{{\sec}^{6}{x}\tan{x}dx}$

$=\displaystyle\int{{\sec}^{5}{x}\sec{x}\tan{x}dx}$

Let $t=\sec{x}\Rightarrow\,dt=\sec{x}\tan{x}dx$

$=\displaystyle\int{{t}^{5}dt}$

We know that $\displaystyle\int{{x}^{n}dx}=\dfrac{{x}^{n+1}}{n+1}+c$

$=\dfrac{1}{6}{t}^{6}+c$

$=\dfrac{1}{6}{\sec}^{6}{x}+c$ where $t=\sec{x}$

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

#### Realted Questions

Q1 One Word Medium
Evaluate$\displaystyle \int_{0}^{\sin ^{2}t}\sin ^{-1}\sqrt{x}dx+\int_{0}^{\cos ^{2}t}\cos ^{-1}\sqrt{x}dx = k$, then $tan(k) =?$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Find the number of values of $x$ satisfing $\int _{ 0 }^{ x }{ { t }^{ 2 }\sin { \left( x-t \right) } dt={ x }^{ 2 } }$in $\left[ 0,100 \right]$y

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle\int _{ 1 }^{ 3 }{ \dfrac { \cos { \left( \log { x } \right) } }{ x } dx }$ is equal to
• A. $1$
• B. $\cos { \left( \log { 3 } \right) }$
• C. $\dfrac { \pi }{ 4 }$
• D. $\sin { \left( \log { 3 } \right) }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
I:$\displaystyle \int\frac{dx}{\sqrt{9-x^{2}}}=\sin^{-1}\left(\frac{x}{3}\right)+c$
II:$\displaystyle \int\frac{\cos x}{\sqrt{16-\sin^{2}x}} dx$-- $\displaystyle \sin^{-1}\left(\frac{\sin x}{4}\right)+c$
• A. Only I
• B. Only II
• C. neither I nor II are true
• D. Both I and II

Evaluate $\int \dfrac{e^x-e^{-x}}{e^x+e^{-x}}dx$