AP EAMCET · Maths · Definite Integration
\(\int_0^2 x^8\left(\frac{4}{x^2}-1\right)^{5 / 2} d x=\)
- A \(\frac{2^{15}}{63}\)
- B \(\frac{2^{16}}{315}\)
- C \(\frac{2^{16}}{189}\)
- D \(\frac{2^{10}}{63}\)
Answer & Solution
Correct Answer
(D) \(\frac{2^{10}}{63}\)
Step-by-step Solution
Detailed explanation
\( \int_0^2 x^8\left(\frac{4}{x^2}-1\right)^{5 / 2} d x = \int_0^2 x^8\left(\frac{4-x^2}{x^2}\right)^{5 / 2} d x = \int_0^2 x^8\frac{(4-x^2)^{5 / 2}}{x^5} d x = \int_0^2 x^3(4-x^2)^{5 / 2} d x \) Let \( u = 4-x^2 \). Then \( du = -2x \, dx \), so \( x \, dx = -\frac{1}{2} du \)…
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