Mathematics

# Evaluate the following integration w.r.t. $x$ $\int \dfrac {1}{(4x+5)^{2}+1}dx$

##### SOLUTION
According to question,

$\int \dfrac{1}{(4x+5)^2+1}dx$

Let $4x+5=t$

$\implies 4dx=dt$

Hence,
$\dfrac{1}{4}\int \dfrac{1}{(t)^2+1}dx$

$\implies \dfrac{1}{4}tan^{-1}t+C$

$\implies \dfrac{1}{4}tan^{-1}(4x+5)+C$

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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
If $2\displaystyle \int_{0}^{1}\tan^{-1}xdx=\displaystyle \int_{0}^{1}\cot^{-1}(1-x+x^{2})dx$, then $\displaystyle \int_{0}^{1}\tan^{-1}(1-x+x^{2})dx$ is equal to:
• A. $\dfrac {\pi}{2}+\log 2$
• B. $\log 2$
• C. $\dfrac {\pi}{2}-\log 4$
• D. $\log 4$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\int{\dfrac{[f(x).\phi (x)-\phi (x).\phi (x)]}{f(x).\phi(x)}\log\dfrac{f(x)}{\phi(x)}dx}$ is equal to
• A. $\dfrac{1}{2}\left[\log\dfrac{\theta(x)}{f(x)}\right]^{2}+k$
• B. $\dfrac{\phi(x)}{f(x)}\log\dfrac{\phi(x)}{f(x)}$
• C. $none\ of\ these$
• D. $\log\dfrac{\phi(x)}{f(x)}+k$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate $\displaystyle \int {\frac{{dx}}{{\sqrt {x + 1} - \sqrt x }}}$.

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
If $f$ is continuous on $R$ and $n\: \epsilon \: N$, then value of $\displaystyle \int_{-3}^{3}x^{2n}\left ( f\left ( x \right )+f\left ( -x \right ) \right )\: dx$ is
• A. $\displaystyle \frac{n}{2}f\left ( 3 \right )$
• B. $\displaystyle \frac{n}{3}f\left ( 3 \right )$
• C. $nf\left ( 3 \right )$
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
Let us consider the integral of the following forms
$f{(x_1,\sqrt{mx^2+nx+p})}^{\tfrac{1}{2}}$
Case I If $m>0$, then put $\sqrt{mx^2+nx+C}=u\pm x\sqrt{m}$
Case II If $p>0$, then put $\sqrt{mx^2+nx+C}=u\pm \sqrt{p}$
Case III If quadratic equation $mx^2+nx+p=0$ has real roots $\alpha$ and $\beta$, then put $\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$