Question

Integrate by using suitable substitution

(1). $\int \sqrt{3-2s}$ ds

(2). $\int \csc \left(\frac{v-\pi }{2}\right)\cot \left(\frac{v-\pi }{2}\right)dv$
(3). $\underset{\pi }{\overset{2\pi }{\int }}\theta d\theta$
(4). $\underset{0}{\overset{\sqrt{x}}{\int }}x\sin x^{2}dx$

Answer 1

$1)\int \sqrt{3-2s}ds=\frac{(3-2s{)}^{\frac{3}{2}}}{(-2).\frac{3}{2}}+c=\frac{-1}{3}(3-2s{)}^{\frac{3}{2}}+c$

$2)\int \csc (\frac{v-\pi }{2})\cot (\frac{v-\pi }{2})dv=2\csc (\frac{v-\pi }{2})+c$

$3){\int }_{\pi }^{2\pi }\theta d\theta =\frac{1}{2}[4{\pi }^2-{\pi }^2]=\frac{3{\pi }^2}{2}$

$4){\int }_0^{\sqrt{x}}x\sin x^{2}dx=\frac{1}{2}[\sin x^2{]}_0^{\sqrt{x}}=\frac{1}{2}\sin x$