The value of \int_{0}^{1} 4 x^{3} \left{\right. \frac{d^{2}}{\left(dx\right)^{2}} \left(\left(\right. 1 - x^{2} \left.\right)\right)^{5} \left.\right} \textrm{ } dxis

Engineering

Mathematics

Properties of Definite Integral

Question

The value of $\int_{0}^{1} 4 x^{3} {\frac{d^{2}}{{dx}^{2}} {(1 - x^{2})}^{5}} dx$ is

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Let $I = \int_{0}^{1} \underset{I}{\underset{⏟}{4 x^{3}}} \underset{II}{\underset{⏟}{f'' (x)}} dx$ where f (x) = (1 – x²)⁵ (using I.B.P.)

$= - 12 \int_{0}^{1} \underset{I}{\underset{⏟}{x^{2}}} \underset{II}{\underset{⏟}{f' (x)}} dx$ (Using I.B.P.)

$= 24 \int_{0}^{1} x {(1 - x^{2})}^{5} dx$

Put 1 – x² = t

$∴$ I = 2