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  • Please solve RD Sharma class 12 chapter Indefinite Integrals exercise 18.16 question 9 maths textbook solution

Please solve RD Sharma class 12 chapter Indefinite Integrals exercise 18.16 question 9 maths textbook solution

Answers (1)

Answer:

        \frac{1}{6 a^{3}} \log \left|\frac{x^{3}-a^{3}}{x^{3}+a^{3}}\right|+C

Hint:

Use substitution method as well as special integration formula to solve this type of problem

Given:

        \int \frac{x^{2}}{x^{6}-a^{6}}dx

Solution:

Let\: \: I=\int \frac{x^{2}}{x^{6}-a^{6}}dx        

                =\int \frac{x^{2}}{(x^{3})^{2}-(a^{3})^{2}}dx

Put\: \: x^{3}=t\Rightarrow 3x^{3}dx=dt\Rightarrow x^{2}dx=\frac{dt}{3}

Then\: \: I=\int \frac{1}{t^{2}-(a^{3})^{2}}\: \: \frac{dt}{2}

                    \begin{aligned} &=\frac{1}{3} \int \frac{1}{t^{2}-\left(a^{3}\right)^{2}} d t \\ &=\frac{1}{3} \times \frac{1}{2\left(a^{3}\right)} \log \left|\frac{t-a^{3}}{t+a^{3}}\right|+C \quad\left[\because \int \frac{1}{x^{2}-a^{2}} d x=\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+C\right] \\ &=\frac{1}{6 a^{3}} \log \left|\frac{x^{3}-a^{3}}{x^{3}+a^{3}}\right|+C \quad\left[\because t=x^{3}\right] \end{aligned}

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Gurleen Kaur

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