Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa

The integral \int \tfrac{ \sin ^{2}x \cos ^{2}x}{(\sin ^{5}x+\cos ^{3}x\sin ^{2}x+\sin ^{_{3}}x\cos ^{2}x+\cos ^{5})^{2}} is equal to : 

  • Option 1)

    \frac{-1}{1+\cot ^{3}x} +\mathbb{C}

  • Option 2)

    \frac{1}{3(1+\tan ^{3}x)} +\mathbb{C}

  • Option 3)

    \frac{-1}{3(1+\tan ^{3}x)} +\mathbb{C}

  • Option 4)

    \frac{1}{1+\cot ^{3}x}+c

 

Answers (2)

best_answer

\int \frac{ \sin ^{2}x \cos^{2}x dx}{(\sin^{3}x (\sin^{2}x + \cos ^{2}x)+ \cos^{2}x (\sin^{2}x + \cos ^{2}x))^{2}}

=\int \frac{ \sin ^{2}x \cos^{2}x dx}{(\sin^{3}x+ \cos^{3}x )^{2}}

=\int \frac{ \sin ^{2}x \cos^{2}x dx}{ \cos^{6}x (1+ \tan ^{3}x)^{2}}

=\int \frac{ \tan ^{2}x \sec^{2}x dx}{ (1+ \tan ^{3}x)^{2}}

put { (1+ \tan ^{3}x)}=t

3 \tan^{2}x\cdot \sec^{2}x dx= dt

I = \int 1/3t^{2}dt = \frac{-1}{t}\times 1/3 + c= -1/3(1+\tan^{3}x)+ c

 

Integration by substitution -

The functions when on substitution of the variable of integration to some quantity gives any one of standard formulas.

 

 

- wherein

Since \int f(x)dx=\int f(t)dt=\int f(\theta )d\theta all variables must be converted into single variable ,\left ( t\, or\ \theta \right )

 

 

 

 

 


Option 1)

\frac{-1}{1+\cot ^{3}x} +\mathbb{C}

This is incorrect

Option 2)

\frac{1}{3(1+\tan ^{3}x)} +\mathbb{C}

This is incorrect

Option 3)

\frac{-1}{3(1+\tan ^{3}x)} +\mathbb{C}

This is correct

Option 4)

\frac{1}{1+\cot ^{3}x}+c

This is incorrect

Posted by

Himanshu

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2024: NTA debars 39 candidates for 3 years on account of using unfair means
anam-khan

JEE Main 2024 session 2 paper 2 answer key objection window closes today; fees
anam-khan

JEE Main 2024 Topper: Farmer’s son Gajare Nilkrishna says ‘keep questioning until you understand’

Latest Question