Mathematics

# Evaluate the integrals using substitution.$\displaystyle\int^1_0\dfrac{x^2}{x^2+1}dx$.

##### SOLUTION
$\int _{ 0 }^{ 12 }{ \cfrac { { x }^{ 2 } }{ { x }^{ 2 }+1 } } dx$
$x=\tan { \theta } \Rightarrow dx=\sec ^{ 2 }{ \theta } d\theta \quad$
$x=0\Rightarrow \theta ={ 0 }^{ o },x=1\Rightarrow \theta =\cfrac { \pi }{ 4 }$
$\quad =\int _{ 0 }^{ \pi /4 }{ \cfrac { \tan ^{ 2 }{ \theta } }{ \sec ^{ 2 }{ \theta } } .\sec ^{ 2 }{ \theta } .d\theta } =\int _{ 0 }^{ \pi /4 }{ \tan ^{ 2 }{ \theta } } .d\theta$
$=\int _{ 0 }^{ \pi /4 }{ \left( \sec ^{ 2 }{ \theta } -1 \right) } .d\theta ={ \left[ \tan { \theta } -\theta \right] }_{ 0 }^{ \pi /4 }=1-\cfrac { \pi }{ 4 }$

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

#### Realted Questions

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Find the value of $n$ such that $\displaystyle\int\limits_{-2}^{2} x^n \ dx=0.$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int e^x\, \left ( \frac{x^2\, -\, 3}{(x\, +\, 3)^2}\right )dx$ equals to
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1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard

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Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020