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

Q1 Subjective Medium
Find the value of $n$ such that $\displaystyle\int\limits_{-2}^{2} x^n \ dx=0.$

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
• A. $\displaystyle e^x\, \frac{x}{x\, +\, 3}\, +\, c$
• B. $\displaystyle e^x\, \left ( 2\, -\, \frac{6}{x\, +\, 3}\right) +\, c$
• C. $\displaystyle e^x\, \frac{3}{x\, +\, 3}\, +\, c$
• D. $\displaystyle e^x\, \left ( 1\, -\, \frac{6}{x\, +\, 3}\right) +\, c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate
$\int {\dfrac{{dx}}{{\left( {x + 1} \right)\left( {x + 5} \right)}}}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle\overset{\pi/6}{\underset{\pi/3}{\displaystyle\int}} \dfrac{1}{1+\sqrt{\tan x}}dx=$ _____________.
• A. $\dfrac{\pi}{12}$
• B. $\dfrac{\pi}{4}$
• C. $\dfrac{\pi}{2}$
• D. $-\dfrac{\pi}{12}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard

In calculating a number of integrals we had to use the method of integration by parts several times in succession. The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts.

$\int u(x)\, v(x)dx\, =\, u(x)\, v_{1}(x)\, -\, u^{}(x)v_{2}(x)\, +\, u^{}(x)\, v_{3}(x)\, -\, .\, +\, (-1)^{n\, -\, 1}u^{n\, -\, 1}(x)v_{n}(x)\, -\, (-1)^{n\, -\, 1}$ $\int\, u^{n}(x)v_{n}(x)\, dx$ where $v_{1}(x)\, =\, \int v(x)dx,\, v_{2}(x)\, =\, \int v_{1}(x)\, dx\, ..\, v_{n}(x)\, =\, \int v_{n\, -\, 1}(x) dx$

Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration  by parts is especially useful when calculating $\int P_{n}(x)\, Q(x)\, dx$, where $P_{n}(x)$, is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n + 1 times.