Solved by the radical integration. \int\frac{\sqrt{1-x^2}}{x^4}dx

Cem Hayes 2021-05-12 Answered
Solved by the radical integration.
\(\int\frac{\sqrt{1-x^2}}{x^4}dx\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Arham Warner
Answered 2021-05-13 Author has 13739 answers

Solution:
\(\int\frac{\sqrt{1-x^2}}{x^4}dx\)
Let \(x=\sin\theta\Rightarrow dx=\cos\theta\cdot d\theta\)
\(=\int\frac{\sqrt{1-\sin^2\theta}}{\sin^4\theta}\cos\theta d\theta\)
\(=\int\frac{\cos^2\theta}{\sin^4\theta}d\theta=\int\cot^2\theta\cdot\csc^2\theta d\theta\)
Let \(u=\cot\theta\Rightarrow du=-\csc^2\theta d\theta\)
\(=-\int u^2 du\)
\(=-\frac{u^3}{3}+c\)
\(=-\frac{\cot^3\theta}{3}+c\)
\(=-\frac{(1-x^2)\sqrt{1-x^2}}{3\times3}+c\)
\(\int\frac{\sqrt{1-x^2}}{x^4}dx=-\frac{(1-x^2)\sqrt{1-x^2}}{3\times3}+c\)

Have a similar question?
Ask An Expert
24
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-08-15
Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.)
\(\displaystyle{7}{{\sin}^{{{8}}}{\left({x}\right)}}{\cos{{\left({x}\right)}}}{\ln{{\left({\sin{{\left({x}\right)}}}\right)}}}{\left.{d}{x}\right.}\)
no. 101. \(\displaystyle\int{u}^{{{n}}}{\ln{{u}}}{d}{u}={\frac{{{u}^{{{n}+{1}}}{\left\lbrace{\left({n}+{1}\right)}^{{{2}}}\right\rbrace}{\left[{\left({n}+{1}\right)}{\ln{{u}}}-{1}\right]}+{C}}}{}}\)
asked 2021-08-17
Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.)
\(\displaystyle\int{37}{e}^{{{74}{x}}}{\arctan{{\left({e}^{{{37}{x}}}\right)}}}{\left.{d}{x}\right.}\)
Inverse Trigonometric Forms (92): \(\displaystyle\int{u}{{\tan}^{{-{1}}}{u}}\ {d}{u}={\frac{{{u}^{{{2}}}+{1}}}{{{2}}}}{{\tan}^{{-{1}}}{u}}-{\frac{{{u}}}{{{2}}}}+{C}\)
asked 2021-05-14
Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.)
\(\int37e^{74x}\arctan(e^{37x})dx\)
Inverse Trigonometric Forms (92): \(\int u\tan^{-1}u\ du=\frac{u^{2}+1}{2}\tan^{-1}u-\frac{u}{2}+C\)
asked 2021-05-02
Evaluate the integral. (Use C for the constant of integration.)
\(\int \ln (\sqrt x)dx\)
asked 2021-08-21
Evaluate integration
\(\displaystyle\int{\frac{{{1}}}{{{\left({x}^{{2}}-{9}\right)}^{{{\frac{{{3}}}{{{2}}}}}}}}}{\left.{d}{x}\right.}\)
using trigonometry substitution
asked 2021-08-14
\(\displaystyle\int{\frac{{\sqrt{{{x}^{{2}}-{1}}}}}{{{x}^{{4}}}}}{\left.{d}{x}\right.}\)
asked 2021-08-18
Evaluate the following
1.\(\displaystyle\int{e}^{{x}}\sqrt{{{1}-{e}^{{{2}{x}}}}}{\left.{d}{x}\right.}\)
2. \(\displaystyle\int{\frac{{{x}^{{2}}}}{{\sqrt{{5}}+{x}^{{2}}}}}{\left.{d}{x}\right.}\)
...