Mathematics

# Integrate $\displaystyle \int {\frac{{{v^2}}}{{{v^2} + 2v + 1}}} dx$

##### SOLUTION
$\int \frac{v^{2}}{v^{2}+2v+1} dv$
$\Rightarrow \int \frac{v^{2} + (2v+1) - (2v+1)}{v^{2} + 2v + 1} dv$
$\Rightarrow \int \frac{(v^{2} + 2v + 1) - (2v+1)}{v^{2}+2v+1} dv = \int dv - \int \frac{2v+1}{v^{2}+2v+1}dv$
$\Rightarrow v-\int \frac{2v+1 +(1-1)}{v^{2}+2v+1} dv$
$\Rightarrow v-\int \frac{2v+2-1)}{v^{2}+2v+1} dv$
$\Rightarrow v-\left [ \int \frac{2v+2dv}{v^{2}+2v+1} - \int \frac{dv}{v^{2}+2v+1} \right ]$
$\Rightarrow v - \left [ \int \frac{(2v+2)dv}{v^{2}+2v+1} - \int \frac{dv}{v^{2}+2v+1} \right ]$
Let $v^{2}+2v+1 = t$
(2v+2)dv =dt
$\Rightarrow v-\left [ \int \frac{dt}{t} - \int \frac{dv}{(v+1)^{2}} \right ]$
$\Rightarrow v- ln\left | t \right |$ $+ \frac{-1}{v+1} + c$
$\Rightarrow v - ln \left | v^{2}+2v+1\right |$ - $\frac{1}{v+1} + c$

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

#### Realted Questions

Q1 Single Correct Medium
Solve  $\displaystyle \int \tan^{-1}xdx$
• A. $x\tan x -\log|1+x^{2}|+c$
• B. $x$  $\tan^{-1} x+\log\sqrt{1+x^{2}}+c$
• C. $x\tan x+\log|1+x^{2}|+c$
• D. $x$ $\tan^{-1} x -\displaystyle \frac{1}{2}\log|1+x^{2}|+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
The value of the integral $\displaystyle \int_{\frac{\pi}6}^{\frac{\pi}2} \left (\dfrac {1 + \sin 2x + \cos 2x}{\sin x + \cos x}\right )dx$ is equal to
• A. $16$
• B. $8$
• C. $4$
• D. $1$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
The value of the integral $\displaystyle \int_{-\pi/4}^{\pi/4}\log (\sec \theta - \tan \theta)d \theta$ is
• A. $\frac{\pi}{4}$
• B. $\frac{\pi}{2}$
• C. $\pi$
• D.

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $\displaystyle f(x)=\frac{{x}^{7}-3{x}^{5}+7{x}^{3}-x+1}{{\cos}^{2}x}$ then $\displaystyle \int_{-\pi/4}^{\pi/4}{f(x)}dx$ is equals to
• A. $\frac{\pi}{2}$
• B. $0$
• C. $\displaystyle \int_{0}^{\pi/4}{f(x)}dx$
• D. $2$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$