Mathematics

# $\displaystyle\int{\dfrac{1+\cos{x}}{x+\sin{x}}dx}$

##### SOLUTION
Let $t=x+\sin{x}\Rightarrow dt=\left(1+\cos{x}\right)dx$
$\displaystyle\int{\dfrac{1+\cos{x}}{x+\sin{x}}dx}$
$=\displaystyle\int{\dfrac{dt}{t}}$
$=\log{t}+c$
$=\log{\left(x+\sin{x}\right)}+c$ where $c$ is the constant of integration.

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

#### Realted Questions

Q1 Single Correct Medium
The value of $\displaystyle \int_{0}^{\pi }\frac{\sin nx}{\sin x}dx$ is?
where $n\epsilon N$
• A. $\pi$ if $n$ is even
• B. $0$ if $n$ is odd
• C. $\displaystyle \pi$ for all $n\displaystyle \epsilon$ N
• D. $0$ if $n$ is even

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
The value of $\int {\frac{{dx}}{{{{\left( {1 - {x^2}} \right)}^{3/2}}}}}$ equals
• A. $x\sqrt {1 - {x^2}} + c$
• B. $\frac{x}{{2\sqrt {1 - {x^2}} }} + c$
• C. $\frac{{2x}}{{\sqrt {1 - {x^2}} }} + c$
• D. $\frac{x}{{\sqrt {1 - {x^2}} }} + c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Show that $\displaystyle \int \frac{x dx}{(px+q)^{3/2}}=\frac{1}{p^{2}}\left \{ \sqrt{px+q}+\frac{q}{\sqrt{(px+q)}} \right \}\cdot$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Integrate with respect to x:
$\sec^2 x\tan x$.
• A. $tan^2x+C$
• B. $sec^2x+C$
• C. $cot^2x+C$
• D. $\dfrac{tan^2x}{2}+C$

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