Mathematics

$$\displaystyle\int { \sqrt { 4-{ x }^{ 2 } }  } dx$$ is equals to


SOLUTION
$$I=\displaystyle\int{\sqrt{4-{x}^{2}}dx}$$

Take $$x=2\sin{\theta}\Rightarrow dx=2\cos{\theta}d{\theta}$$

$$\sqrt{4-{x}^{2}}=\sqrt{4-4{\sin}^{2}{\theta}}=\sqrt{4\left(1-{\sin}^{2}{\theta}\right)}=2\sqrt{{\cos}^{2}{\theta}}=2\cos{\theta}$$

$$\displaystyle\int{\sqrt{4-{x}^{2}}dx}$$

$$=\displaystyle\int{2\cos{\theta}.2\cos{\theta}d{\theta}}$$

$$=2 \displaystyle \int{2{\cos}^{2}{\theta}d{\theta}}$$

$$=2 \displaystyle \int{\left(1+\cos{2\theta}\right)d{\theta}}$$ since $$1+\cos{2\theta}=2{\cos}^{2}{\theta}$$

$$=2\left(\theta+\dfrac{\sin{2\theta}}{2}\right)+c$$

$$=2\left({\sin}^{-1}{x/2}+\dfrac{\sin{2{\sin}^{-1}{x/2}}}{2}\right)+c$$ since $$x=2\sin{\theta}\Rightarrow \theta={\sin}^{-1}{x/2}$$
View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84
Enroll Now For FREE

Realted Questions

Q1 Single Correct Medium
$$\int_{0}^{\pi /2}sin2xtan^{-1}\left ( sinx \right )dx=$$

  • A.  $$\dfrac{\pi }{2}$$+1
  • B.  $$\dfrac{3\pi }{2}$$+1
  • C.  $$\dfrac{3\pi }{2}$$-1
  • D.  $$\dfrac{\pi }{2}$$-1

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
If $$f(y)=e^y, g(y)-y; y > 0$$ and $$F(t)=\displaystyle\int^t_0f(t-y)g(y)dy$$, then?
  • A. $$F(t)=t e^t$$
  • B. $$F(t)=t e^{-t}$$
  • C. $$F(t)=1-e^{-t}(1+t)$$
  • D. $$F(t)=e^t-(1+t)$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Find $$\int \dfrac {x^{3} - 1}{x^{2}}dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Hard
Evaluate: $$\displaystyle \int \dfrac{\left ( \sec x\:co\sec   \right )}{\left ( \log \tan x \right )} dx$$
  • A. $$\displaystyle \log \left ( \tan x \right )$$
  • B. $$\displaystyle \cot \left ( \log x \right )$$
  • C. $$\displaystyle \tan \left ( \log x \right )$$
  • D. $$\displaystyle \log \log \left ( \tan x \right )$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
Consider two differentiable functions $$f(x), g(x)$$ satisfying $$\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$$ & $$\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$$. where $$\displaystyle f(x)>0    \forall  x \in  R$$

On the basis of above information, answer the following questions :

Asked in: Mathematics - Limits and Derivatives


1 Verified Answer | Published on 17th 08, 2020

View Answer