_{-2}² x^4 (4-x²)^{7/2} dx = [TG EAPCET 9 May 2024 Shift 1]

Correct Answer is : 28π28 π28π

Correct Answer is : 28π28 π28π

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TS EAMCET 9 May 2024 Shift 1 Solved Paper

Section: Mathematics

Question : 77 of 160

Marks: +1, -0

\int\limits_{-2}^{2} x^4 (4-x^2)^{\frac{7}{2}} dx =

Solution:

\;\; \text{ Let } \; I=\int\limits_{-2}^{2} x^4 (4-x^2)^{\frac{7}{2}} dx

\;I=2 \int\limits_{0}^{2} x^4 (4-x^2)^{\frac{7}{2}} dx

\;\;\; [ \because f(x) \; \text{ is even function } \;]

\;\; \text{ put } \; x=2 \sin \theta \Rightarrow dx=2 \cos \theta \, d\theta

\;\; \text{ when } \; x=0, \theta=0 \; \text{ and } \; x=2, \theta=\frac{\pi}{2}

\; \therefore I=2 \int\limits_{0}^{\pi/2} 2^4 \sin^4 \theta (2 \cos \theta)^7 \cdot 2 \cos \theta \, d\theta

\;I=2 \cdot 2^4 \cdot 2^8 \int\limits_{0}^{\pi/2} \sin^4 \theta \cos^8 \theta \, d\theta

\;I=2^{13} .

\;\; \frac{(4-1)(4-3)(8-1)(8-3)(8-5)(8-7)}{12 \times 10 \times 8 \times 6 \times 4 \times 2} \times \frac{\pi}{2}

(by Wall's formulae)

\;=\; \frac{2^{13} \times 3 \times 1 \times 7 \times 5 \times 3 \times 1 \times \frac{\pi}{2}}{12 \times 10 \times 8 \times 6 \times 4 \times 2}

\;=\; \frac{2^6 \times 3 \times 7 \times 5 \times 3 \times \pi}{6 \times 5 \times 4 \times 3 \times 2}=28 \pi

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