Question

50 points each question. Please help. How do I solve?

Answer 1

`I=\displaystyle \int ^(\pi)_{\tfrac{\pi}3} ( \sin x)/(1 + \cos^2 x) dx\\ \\\\\text{let,}\\\\~~~~~u=\cos x\\\\\implies (du)/(dx) =-\sin x\\ \\\implies \sin x~~ dx = -du\\\\\text{When}~~ x = \pi , ~~ u = \cos \pi = -1\\\\\text{When}~~ x = \frac{\pi}3 , ~~u = \cos \frac{\pi}3 =\frac 12\\ \\\\I =- \displaystyle \int ^(-1)_(\tfrac 12) (du)/(1+u^2)\\\\\\`

`=\displaystyle \int ^(\tfrac 12)_(-1) (du)/(1+u^2)~~~~~~~~~~;\left[\displaystyle \int^(a)_b f(x) dx = - \displaystyle \int^(b)_a f(x) dx ,~ b < a\right]\\\\\\=\left[\tan^(-1) u \right]^(\tfrac 12)_(-1)~~~~~~~~;\left[ \ddisplaystyle \int (dx)/( 1+ x^2) = \tan^(-1) x + C \right]\\\\\\=\tan^(-1) \left( \frac 12 \right) + \tan^(-1) 1\\\\\\=\tan^(-1) \left( \frac 12 \right) + \frac{\pi}4 \\\\\\=1.249`