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Please solve the attached problem in integral calculus.

Please solve the attached problem in integral calculus.

Question Image
Grade:12

1 Answers

Sami Ullah
46 Points
2 years ago
Its very easy just a few manipulations you’ll have to do.First I will just solve the integral and later apply the upper and lower limits.
 
I=\int\frac{\sqrt{x}}{\sqrt{1-x^3}}.dx
 
Let            u=x^\frac{3}{2}    \Rightarrow      u^2=x^3
 
Then       du=\frac{3}{2}x^\frac{1}{2}.dx
 
\Rightarrow            \frac{2}{3}du=\sqrt{x}.dx
 
Now  substituting  values,
 
I=\int \frac{2}{3}\times \frac{1}{\sqrt{1-u^2}}.du
 
I= \frac{2}{3}\sin^{-1}(u)
 
I= \frac{2}{3}\sin^{-1}(x^\frac{3}{2})
 
Putting values,
 
I= \frac{2}{3}[\sin^{-1}(1^\frac{3}{2})-\sin^{-1}(0)]
 
I= \frac{2}{3}[\frac{\pi }{2}-0]
 
I= \frac{\pi }{3}
 
So the awnser is option (2).
 
     
 
 

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