Mathematics

# $\displaystyle \int \sqrt{1-\sin x} \ dx$

##### SOLUTION
$I= \displaystyle\int \sqrt{1-\sin x}dx$

We have $\sqrt{1-\sin x}=\sqrt{\sin^{2}\dfrac{x}{2}+\cos^{2}\dfrac{x}{2}-2 \sin\dfrac{x}{2}\cos\dfrac{x}{2}}$

$=\sqrt{\left(\sin\dfrac{x}{2}-\cos\dfrac{x}{2}\right)^{2}}$

$=\sin\dfrac{x}{2}-\cos\dfrac{x}{2}$

$\therefore I= \displaystyle\int \left(\sin\dfrac{x}{2}-\cos\dfrac{x}{2}\right)dx$

$= \displaystyle\int \sin\dfrac{x}{2}dx-\int \cos\dfrac{x}{2}dx$

$=-\dfrac{\cos\dfrac{x}{2}}{\dfrac{1}{2}}-\dfrac{\sin\dfrac{x}{2}}{\dfrac{1}{2}}=-2\left[\cos\dfrac{x}{2}+\sin\dfrac{x}{2}\right]$

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 Single Correct Hard
Solve: $\displaystyle\int \dfrac {dx}{\sqrt {21-4x-x^2}}$
• A. $\arcsin \dfrac{(x+4)}{5} +C$
• B. $\arcsin \dfrac{(x-4)}{5} +C$
• C. None of these
• D. $\arcsin \dfrac{(x+2)}{5} +C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Prove that $\displaystyle \int \sqrt{x^2 - a^2} dx = \dfrac{x}{2} \sqrt{x^2 - a^2} - \dfrac{a^2}{2} \log |x + \sqrt{x^2 - a^2}| + c$.

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate $\displaystyle \int { \frac { 1 }{(1-2x) } }dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Integrate the function   $\displaystyle \frac {x}{9-4x^2}$

If $y=2^23^{2x}5^{-5}7^{-5}$ then $\dfrac{dy}{dx}=$