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  • If  <img alt="\int f(x)dx=\Psi (x),then\; \int x^{5}f(x^{3})dx" src="https://learn.careers360.com/latex-image/?%5Cdpi%7B100%7D%20%5Cint%20f%28x%29dx%3D%5CPsi%20%28x%29%2Cthen%5C%3B%20

If  \int f(x)dx=\Psi (x),then\; \int x^{5}f(x^{3})dx   is equal to :

Option: 1

\frac{1}{3}\left [ x^{3}\Psi \left ( x^{3} \right )-\int x^{3}\Psi (x^{3})dx \right ]+C


Option: 2

\frac{1}{3}\left [ x^{3}\Psi \left ( x^{3} \right )-\int x^{2}\Psi (x^{3})dx \right ]+C


Option: 3

\frac{1}{3}\, \, x^{3}\Psi \left ( x^{3} \right )-3\int x^{3}\Psi (x^{3})dx +C


Option: 4

\frac{1}{3}\, \, x^{3}\Psi \left ( x^{3} \right )-\int x^{2}\Psi (x^{3})dx +C


Answers (1)

best_answer

As learnt in concept

Integration By PARTS -

Let u and v are two functions then 

\int u\cdot vdx=u\int vdx-\int \left ( \frac{du}{dx}\int vdx \right )dx

- wherein

Where u is the Ist function v is he IInd function

 

\int f(x)dx=\Psi (x)\\ \int x^{5}f(x^{3})dx= \frac{1}{3}\int (3x^{2}).x^{3}f(x^{3})dx\\ Put \:x=t;3x^{2}dx= dt\\ 

\frac{1}{3}\int tf(t)dt=\frac{1}{3}\left [ t\int f(t)dt-\int \int f(t)dt \right ]

Replace t by x3

\frac{1}{3}x^{3}\Psi (x^{3})-\frac{1}{3}\int \Psi(x^{3})dt\\ \frac{1}{3}x^{3}\Psi (x^{3})-\frac{1}{3}\int 3x^{2}dx.\Psi (x^{3})\\ \left [ \frac{1}{3} x^{3}\Psi (x^{3})-\int x^{2}\Psi (x^{3})dx\right ]+c

Posted by

himanshu.meshram

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