Mathematics

# $\int \frac{1}{x(x^n+1)}dx$.

##### SOLUTION

$\int(\frac{1}{x(x^n+1)})dx\\=\int(\frac{x^{n-1}}{x^n(x^n+1)})dx\\let\>x^n=t\>\\nx^{n-1}dx=dt\>\\\therefore\>x^{n-1}dx=(\frac{dt}{n})\\\therefore\>I=(\frac{1}{n})\int\>(\frac{1}{t(t+1)}dt)\\=(\frac{1}{n})[\int(\frac{1}{t})dt-\int\>(\frac{1}{t+1})dt]\\=(\frac{1}{n})[logt-log(t+1)]+c\\=(\frac{1}{n})log((\frac{1}{n}))+c\\=\>(\frac{1}{n})log((\frac{x^n}{x^n+1}))+c$

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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
Solve:
$\int{\displaystyle \dfrac {dx}{x(x^{n}-t)}}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Evaluate : $\displaystyle \int\frac{\cos x}{\sin(x-\dfrac{\pi}{6})\sin(x+\dfrac{\pi}{6})}dx$
• A. $\log \left | \displaystyle \frac {2\sin x+1}{2\sin x-1} \right |+C$
• B. $\log \left | \displaystyle \frac {\sin x+1}{\sin x-1} \right |+C$
• C. $\log \left | \displaystyle \frac {\sin x-1}{\sin x+1} \right |+C$
• D. $\log \left | \displaystyle \frac {2\sin x-1}{2\sin x+1} \right |+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
The integral $\displaystyle \int \frac {dx}{x^2(x^4+1)^{\frac {3}{4}}}$ equals
• A. $\displaystyle \left (\frac {x^4+1}{x^4}\right )^{\frac {1}{4}}+c$
• B. $\displaystyle (x^4+1)^{\frac {1}{4}}+c$
• C. $\displaystyle -(x^4+1)^{\frac {1}{4}}+c$
• D. $\displaystyle -\left (\frac {x^4+1}{x^4}\right )^{\frac {1}{4}}+c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\displaystyle \int \dfrac{\sec^2 x}{\sec x+\tan x)^5}dx=$

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 $\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$