Mathematics

# Evaluate the following integral:$\int { \cfrac { 1+\sin { x } }{ \sqrt { x-\cos { x } } } } dx$

##### SOLUTION
Let
$t=x-\cos{x}\Rightarrow\,dt=\left(1+\sin{x}\right)dx$

$\displaystyle\int{\dfrac{\left(1+\sin{x}\right)dx}{\sqrt{x-\cos{x}}}}$

$=\displaystyle\int{\dfrac{dt}{\sqrt{t}}}$

$=\displaystyle\int{{t}^{\frac{-1}{2}}dt}$

$=\dfrac{{t}^{\frac{-1}{2}+1}}{\dfrac{-1}{2}+1}+c$

$=\dfrac{{t}^{\frac{1}{2}}}{\dfrac{1}{2}}+c$

$=2\sqrt{t}+c$

$=2\sqrt{x-\cos{x}}+c$ where $t=x-\cos{x}$

Its FREE, you're just one step away

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

#### Realted Questions

Q1 Subjective Medium
$\int {{e^x}\left( {{{\sec }^2}x + \tan x} \right)} dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Evaluate:
$\int { \cfrac { 2 }{ (1-x)(1+{ x }^{ 2 }) } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle \int \frac{e^{x}\left ( 1+\sin x \right )}{\left ( 1+\cos x \right )}$dx is equal to
• A. $\displaystyle \log \tan x$
• B. $\displaystyle \sin \log x$
• C. $\displaystyle e^{x}\cos x$
• D. $\displaystyle e^{x}\tan \left ( x/2 \right )$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Find the integral of $\dfrac{1}{\sqrt{x^2-a^2}}$ with respect to $x$ and hence evaluate
$\displaystyle \int{\dfrac{dx}{\sqrt{x^2+6x-7}}}$.

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.