# ∫ x 3 ( log x ) 2 dx - Mathematics

Sum
$\int x^3 \left( \log x \right)^2\text{ dx }$

#### Solution

$\int {x^3}_{II} \cdot \left( \log_I x \right)^2 \cdot dx$
$= \left( \log x^2 \right)\int x^3 dx - \int\frac{2 \log x}{x} \times \frac{x^4}{4} \text{ dx}$
$= \left( \log x \right)^2 \times \frac{x^4}{4} - \frac{1}{2}\int \log_I x \cdot {x^3}_{II} \text{ dx }$
$= \left( \log x \right)^2 \times \frac{x^4}{4} - \frac{1}{2}\left[ \log x\int x^3 dx - \int\left\{ \frac{d}{dx}\left( \log x \right)\int x^3 dx \right\}dx \right]$
$= \left( \log x \right)^2 \times \frac{x^4}{4} - \frac{1}{2} \left[ \log x \cdot \frac{x^4}{4} - \int\frac{1}{x} \times \frac{x^4}{4}dx \right]$
$= \left( \log x \right)^2 \times \frac{x^4}{4} - \frac{1}{2} \left[ \log x \cdot \frac{x^4}{4} - \frac{1}{4}\int x^3 dx \right]$
$= \left( \log x \right)^2 \times \frac{x^4}{4} - \frac{1}{2} \left[ \log x \cdot \frac{x^4}{4} - \frac{x^4}{16} \right] + C$
$= \left( \log x \right)^2 \times \frac{x^4}{4} - \frac{\log x \cdot x^4}{8} + \frac{x^4}{32} + C$

Concept: Indefinite Integral Problems
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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Revision Excercise | Q 100 | Page 204