Mathematics

Integrate $$\int {\sqrt {1 - {t^2}} } dt$$.


SOLUTION
$$ \displaystyle \int \sqrt{1-t^{2}} dt $$

let $$ t = sin\theta \Rightarrow \theta = sin^{-1}(t) $$ and $$ cos\theta =\sqrt{1-sin^{2}\theta}$$

$$ = \sqrt{1-t^{2}}$$
$$ dt = cos\theta d\theta$$

$$ \displaystyle\int \sqrt{1-sin^{2}\theta} \,\,cos\theta d\theta = \int cos\theta . cos\theta .d\theta$$
$$ \displaystyle= \int cos^{2}\theta d\theta$$

$$  \displaystyle  = \int (\dfrac{1+cos2\theta}{2}) d\theta$$
$$ \displaystyle= \dfrac{1}{2}\int d\theta+\dfrac{1}{2}\int cos2\theta d\theta $$

$$ \displaystyle = \dfrac{1}{2}\theta + \dfrac{1}{2}.\dfrac{sin2\theta}{2}+C$$
$$\displaystyle  = \dfrac{\theta}{2}+\dfrac{sin2\theta}{4}+c$$

$$ \displaystyle = \dfrac{1}{2}sin^{-1}(t)+\dfrac{2sincos\theta}{4}+c$$

$$\displaystyle  = \dfrac{1}{2}sin^{-1}(t)+\dfrac{1}{2}.t\sqrt{1-t^{2}}+c$$

$$ \displaystyle \int \sqrt{1-t^{2}}dt = \dfrac{1}{2} sin^{-1}(t)+\frac{t}{2}\sqrt{1-t^{2}}+c$$         (where $$t=\sin\theta$$)
View Full Answer

Its FREE, you're just one step away

Create your Digital Resume For FREE on your name's sub domain "yourname.wcard.io". Register Here!


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

Realted Questions

Q1 Subjective Hard
Prove that $$\displaystyle\int _{ 0 }^{ { \pi  }/{ 4 } }{ 2 } { \tan }^{ 3 }x dx =1-\log{2}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Hard

  Integrate
by using a suitable substitution:-

 (a) $$\int {{{\rm{3}} \over {{{\left( {{\rm{2 - x}}} \right)}^{\rm{2}}}}}{\rm{dx}}} $$

(b) $${\rm{\;\;}}\int {{\rm{sin}}\left( {{\rm{8z - 5}}} \right){\rm{dz}}} $$


Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Medium
If $$f(x)={ \{  }_{ 0, \quad \quad \quad |x|>1\quad \quad \quad \quad \quad \quad \quad \quad \quad  }^{ 1-|x|,\quad \quad |x|<1 }$$ and $$g(x)=f(x-1)+f(x+1)$$ then $$\int _{ 0 }^{ 3 }{ g(x)dx } $$ is equal to
  • A. 1
  • B. 2
  • C. 3
  • D.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
 $$\displaystyle \int \cot x {dx}= $$
  • A. $$\ln (\sin^2x) +C$$
  • B. $$ (\sin x) +C$$
  • C. None of these
  • D. $$\ln (\sin x) +C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Single Correct Medium
$$\displaystyle \int\sqrt{\frac{\cos x-\cos^{3}x}{1-\cos^{3}x}}dx=$$
  • A. $$\displaystyle \frac{2}{3}\sin^{-1}(\cos^{\frac{3}{2}}x)+c$$
  • B. $$\displaystyle \frac{3}{2}\sin^{-1}(\cos^{\frac{3}{2}}x)+c$$
  • C. $$\displaystyle \frac{3}{2} \cos^{-1} (\cos^{\frac{3}{2}}x)+c$$
  • D. $$\displaystyle \frac{2}{3} \cos^{-1} (\cos^{\frac{3}{2}}x)+c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer