Mathematics

# Evaluate$\displaystyle\int\sqrt { 1-{ x }^{ 2 } dx}$

##### SOLUTION
$\displaystyle\int{\sqrt{1-{x}^{2}}dx}$
Let $x=\sin{\theta}\Rightarrow dx=\cos{\theta}d\theta$
$=\displaystyle\int{\sqrt{1-{\sin}^{2}{\theta}}\cos{\theta}d\theta}$
$=\displaystyle\int{\sqrt{{\cos}^{2}{\theta}}\cos{\theta}d\theta}$
$=\displaystyle\int{{\cos}^{2}{\theta}d\theta}$
$=\displaystyle\int{{\cos}^{2}{\theta}d\theta}$ since $\cos{2\theta}=2{\cos}^{2}{\theta}-1\Rightarrow {\cos}^{2}{\theta}=\dfrac{1}{2}\left(1+\cos{2\theta}\right)$
$=\dfrac{1}{2}\displaystyle\int{\left(1+\cos{2\theta}\right)d\theta}$
$=\dfrac{1}{2}\left[\theta+\dfrac{\sin{2\theta}}{2}\right]+c$
$=\dfrac{{\sin}^{-1}{x}}{2}+\dfrac{\sin{2\left({\sin}^{-1}{x}\right)}}{4}+c$ where $\theta={\sin}^{-1}{x}$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Subjective Medium
Evaluate the following integral as the limit of sum:
$\displaystyle\int^2_0e^xdx$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
$\int {\dfrac{{{e^x}dx}}{{\left( {1 + {e^{2x}}} \right)}}}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Prove that $\displaystyle\int^1_{-1}e^{|x|}dx=2(e-1)$.

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \int_0^{\pi / 2} \dfrac{1}{1 + \sqrt [ 4]{tanx }}dx =$
• A. $\pi / 3$
• B.
• C. None of these
• D. $\pi / 4$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Hard
If $\displaystyle I= \int_{0}^{1}\frac{x dx}{8+x^{3}}$ then the smallest interval in which $I$ lies is
• A. $\displaystyle \left ( 0,\frac{1}{8} \right )$
• B. $\displaystyle \left ( 0,\frac{1}{10} \right )$
• C. $\displaystyle \left ( 0,\frac{1}{7} \right )$
• D. $\displaystyle \left ( 0,\frac{1}{9} \right )$