Mathematics

Integrate with respect to $$x$$:
$$\sec^{3}x$$


SOLUTION
$$I= \displaystyle \int\sec^3x\,\,dx$$
$$=\displaystyle \int\sec x. \, \, sec ^2x\,\,dx$$
using formula of integration by parts:
$$I=\sec x. \displaystyle \int\sec ^2dx-\displaystyle \int(\sec \,x)^1(\displaystyle \int\sec ^2x\,\,dx)$$
$$=\sec \, x.tan \,\, x-\displaystyle \int(\sec \, x \tan \, x)(\tan \, x)dx$$
$$\sec\, x \tan \, x- \displaystyle \int \sec \,x.\tan ^2xdx$$
$$\sec \,x \tan \,x - \displaystyle \int\sec \,x (\sec^2x-1)dx$$
$$\sec \,\, x. \tan \, x - \displaystyle \int\sec ^3xdx+ \displaystyle \int \sec \, x\, dx$$
$$I=\sec \, x .\tan \, x -I+ |n|\sec \,x + \tan \, x |+C$$
$$\Rightarrow 2 I = \sec \, x . \tan \, x + |n|\sec \, x+ \tan \, x|+C$$
$$\Rightarrow \boxed {I=\dfrac{1}{2}\sec \, x. \tan + \dfrac{1}{2}|n|\sec \, x+ \tan \, x|+C}$$
Identities used:
$$\displaystyle \int f(x) g(x) dx=f(x)\displaystyle \int g(x)-\displaystyle \int f(x)\left[\displaystyle \int g(x)dx\right] dx$$
$$\displaystyle \int\sec ^2xdx- \tan \,x +C$$
$$\displaystyle \int\sec x dx - In|\sec \, x+ \tan \, x|+C$$
View Full Answer

Its FREE, you're just one step away


Subjective Hard Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
$$\int \sqrt { \sec x - 1 } d x$$ is equal to

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
Solve $$\int { \left( { x }^{ 6 }+7{ x }^{ 5 }+6{ x }^{ 4 }+5{ x }^{ 3 }+4{ x }^{ 2 }+3x+1 \right)  } { e }^{ x }dx=....+c$$
  • A. $$\sum _{ i=0 }^{ 7 }{ { x }^{ i }{ e }^{ x } } $$
  • B. $$\sum _{ i=0 }^{ 6 }{ i{ e }^{ x } } $$
  • C. $$\sum _{ i=0 }^{ 6 }{ { (ex) }^{ i } } $$
  • D. $$\sum _{ i=0 }^{ 6 }{ { x }^{ i }{ e }^{ x } } $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Evaluate the following integral:
$$\displaystyle\int_{-4}^{4}|x+2|\ dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 One Word Medium
$$\displaystyle \int \dfrac{e^{-x}}{1+ e^{-x}}dx$$ $$=$$  $$-p \log (1+e^{-x})+C$$ then p =

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
If $$y=2^23^{2x}5^{-5}7^{-5}$$ then $$\dfrac{dy}{dx}=$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer