Question-117963

Question Number 117963 by peter frank last updated on 14/Oct/20

Answered by john santu last updated on 14/Oct/20

\int_{- 2}^{2} (x^{3} \cos (\frac{x}{2}) \sqrt{4 - x^{2}}) d x = 0

t h e n \int_{- 2}^{2} \frac{1}{2} \sqrt{4 - x^{2}} d x = \int_{0}^{2} \sqrt{4 - x^{2}} d x

= \frac{1}{4} \times 4 π = π

Answered by Dwaipayan Shikari last updated on 14/Oct/20

\int_{- 2}^{2} (x^{3} c o s \frac{x}{2} + \frac{1}{2}) \sqrt{4 - x^{2}} d x = I = \int_{- 2}^{2} (- x^{3} c o s \frac{x}{2} + \frac{1}{2}) \sqrt{4 - x^{2}}

2 I = \int_{- 2}^{2} \sqrt{4 - x^{2}} (I n t e g r a l i s s y m m e t r i c)

\int \sqrt{4 - x^{2}} d x = \int 2 c o s θ \sqrt{4 - 4 {s i n}^{2} θ} d θ = 4 \int {c o s}^{2} θ

= 2 \int 1 + c o s 2 x = 2 θ + s i n 2 θ = 2 {s i n}^{- 1} \frac{x}{2} + s i n (2 {s i n}^{- 1} \frac{x}{2})

S o \int_{- 2}^{2} \sqrt{4 - x^{2}} d x = 2 π

2 I = 2 π

I = π