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  • Explain solution RD Sharma class 12 chapter Indefinite Integrals exercise 18.16 question 8 maths

Explain solution RD Sharma class 12 chapter Indefinite Integrals exercise 18.16 question 8 maths

Answers (1)

Answer:

            \frac{1}{2}tan^{-1}(x^{6})+C

Hint:

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

Given:

            \int \frac{3x^{5}}{1+x^{12}}dx

Solution:

Let\: \: I=\int \frac{3x^{5}}{1+x^{12}}dx

             =\int \frac{3x^{5}}{1+(x^{6})^{2}}dx

Put\: \: x^{6}=t\Rightarrow 6x^{5}dx=dt

Then\: \: I=\int \frac{3x^{5}}{1+t^{2}}\: \frac{dt}{6x^{5}}

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

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Gurleen Kaur

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