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Provide Solution For  R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise  Revision Exercise Question 103  Maths Textbook Solution.

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Answer: \frac{2}{9}\left(1+x^{3}\right)^{3 / 2}-\frac{2}{3} \sqrt{1+x^{3}}+c

Hint: to solve this question we have to use substitute method

Given:  \int \frac{x^{5}}{\sqrt{1+x^{3}}} d x


\text { Let } I=\int \frac{x^{3} \cdot x^{2}}{\sqrt{1+x^{3}}} d x

\text { put } 1+x^{3}=t^{2}, \text { differentiate on both sides, }

3 x^{2} d x=2 t d t

I=\int \frac{\left(t^{2}-1\right)}{t} \cdot \frac{2 t d t}{3}

I=\frac{2}{3} \int\left(t^{2}-1\right) d t

I=\frac{2}{3} \int t^{2} d t-\frac{2}{3} \int d t

I=\frac{2}{3} \frac{t^{3}}{3}-\frac{2}{3} t+c

I=\frac{2}{9}\left(\sqrt{1+x^{3}}\right)^{3}-\frac{2}{3} \sqrt{1+x^{3}}+c

I=\frac{2}{9}\left(1+x^{3}\right)^{3 / 2}-\frac{2}{3} \sqrt{1+x^{3}}+c




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