calculate-pi-4-pi-3-sinx-1-sin-2-x-dx-

Question Number 51989 by maxmathsup by imad last updated on 01/Jan/19

c a l c u l a t e \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{s i n x}{1 + {s i n}^{2} x} d x

Answered by peter frank last updated on 01/Jan/19

\int \frac{\sin x}{2 - \cos^{2} x} d x

u = \cos x

d u = - \sin x d x

d x = - \frac{d u}{\sin x}

\int \frac{\sin x}{2 - u^{2}} . - \frac{d u}{\sin x}

\int - \frac{d u}{2 - u^{2}}

- \frac{1}{2 \sqrt{2}} [\int \frac{d u}{\sqrt{2} + u} - \int \frac{d u}{\sqrt{2} - u}]

- \frac{1}{2 \sqrt{2}} l n [\frac{\sqrt{2} + u}{\sqrt{2} - u}] + B

- \frac{1}{2 \sqrt{2}} l n [\frac{\sqrt{2} + \cos x}{\sqrt{2} - \cos x}] + B