AP EAMCET · Maths · Indefinite Integration
\(\int x^3(\log x)^2 d x=\)
- A \((\log x)^2 \frac{x^4}{4}+\frac{1}{2}\left[(\log x) \frac{x^4}{4}+\frac{x^4}{16}\right]+C\)
- B \((\log x)^2 \frac{x^4}{4}-\frac{1}{2}\left[(\log x) \frac{x^4}{4}+\frac{x^4}{16}\right]+C\)
- C \((\log x)^2 \frac{x^4}{4}-\frac{1}{2}\left[(\log x) \frac{x^4}{4}-\frac{x^4}{16}\right]+C\)
- D \((\log x)^2 \frac{x^4}{4}+\frac{1}{2}\left[(\log x) \frac{x^4}{4}-\frac{x^4}{16}\right]+C\)
Answer & Solution
Correct Answer
(C) \((\log x)^2 \frac{x^4}{4}-\frac{1}{2}\left[(\log x) \frac{x^4}{4}-\frac{x^4}{16}\right]+C\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text {Let } I=\int x^3(\log x)^2 d x \\ & =(\log x)^2 \cdot \frac{x^4}{4}-\int \frac{2}{4}(\log x) \cdot x^3 d x \\ & =(\log x)^2 \cdot \frac{x^4}{4}-\frac{1}{2}\left[(\log x) \cdot \frac{x^4}{4}-\int \frac{x^3}{4} d x\right] \\ & =(\log x)^2…
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