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  • Provide Solution For R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 100 Maths Textbook Solution.

Provide Solution For  R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise  Revision Exercise Question 100  Maths Textbook Solution.

Answers (1)

Answer:

(\log x)^{2} \times \frac{x^{4}}{4}-\frac{\log x \cdot x^{4}}{8}+\frac{x^{4}}{32}+C

Hint:

You must know about the log formula of integration.

Given:

\int x^{3}(\log x)^{2} d x

Solution:

\int x^{3}{ }_{11} \cdot\left(\log _{1} x\right)^{2} \cdot d x

=\left(\log x^{2}\right) \int x^{3} d x-\int \frac{2 \log x}{x} \times \frac{x^{4}}{4} d x

=(\log x)^{2} \times \frac{x^{4}}{4}-\frac{1}{2} \int \log _{1} x \cdot x^{3}{ }_{I I} d x

=(\log x)^{2} \times \frac{x^{4}}{4}-\frac{1}{2}\left[\log x \int x^{3} d x-\int\left\{\frac{d}{d x}(\log x) \int x^{3} d x\right\} d x\right]

=(\log x)^{2} \times \frac{x^{4}}{4}-\frac{1}{2}\left[\log x \cdot \frac{x^{4}}{4}-\int \frac{1}{x} \times \frac{x^{4}}{4} d x\right]

=(\log x)^{2} \times \frac{x^{4}}{4}-\frac{1}{2}\left[\log x \cdot \frac{x^{4}}{4}-\frac{1}{4} \int x^{3} d x\right]

=(\log x)^{2} \times \frac{x^{4}}{4}-\frac{1}{2}\left[\log x \cdot \frac{x^{4}}{4}-\frac{x^{4}}{16}\right]+C

=(\log x)^{2} \times \frac{x^{4}}{4}-\frac{\log x \cdot x^{4}}{8}+\frac{x^{4}}{32}+C

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