Question

$\int \frac{x}{(7x-10-x^2{)}^{3/2}}dx$

Answer 1

$\int \frac{x}{{(7x-10-x^2)}^{3/2}}dx$

$\Rightarrow \int \frac{\frac{7}{2}-\frac{1}{2}(7-2x)}{{(7x-10-x^2)}^{3/2}}dx$

$\Rightarrow \frac{-1}{2}\int \frac{7-2x}{{(7x-10-x^2)}^{3/2}}+\frac{7}{2}\int \frac{dx}{{(7x-10-x^2)}^{3/2}}$

$u=7x-10-x^2$

$du=(7-2x)dx$

$\frac{-1}{2}\int u^{-3/2}du+\frac{7}{2}\int \frac{dx}{{(\frac{9}{4}-{(x-\frac{7}{2})}^2)}^{3/2}}$

$\Rightarrow \frac{-1}{2}\times (-2)\times u^{-1/2}+\frac{7}{2}\times 8\int \frac{dx}{{(9-{(2x-7)}^2)}^{3/2}}$

$u=2x-7$

$du=2dx$

$\Rightarrow u^{-1/2}+\frac{28}{2}\int \frac{dx}{{(9-u^2)}^{3/2}}$

$\Rightarrow {(7x-10-x^2)}^{-3/2}+14\int \frac{dx}{{(9-u^2)}^{3/2}}$

$u=3\sin \theta$

$du=3\cos \theta d\theta$

$\Rightarrow {(7x-10-x^2)}^{-1/2}+\frac{14}{27}\int \frac{d\theta \times 3\cos \theta }{{(9-{9\sin }^2\theta )}^{3/2}}$

$\Rightarrow {(7x-10-x^2)}^{-1/2}+\frac{14}{9}\int \frac{d\theta }{{\cos }^2\theta }$

$\Rightarrow {(7x-10-x^2)}^{-1/2}+\frac{14}{9}\int {\sec }^{2}d\theta$

$\Rightarrow {(7x-10-x^2)}^{-1/2}+\frac{14}{9}\tan \theta d\theta +c$

$\Rightarrow {(7x-10-x^2)}^{-1/2}+\frac{14}{9}\frac{4}{\sqrt{1-\frac{u^2}{9}}}+c$

$\Rightarrow {(7x-10-x^2)}^{-1/2}+\frac{14}{9}\frac{2x}{\sqrt{1-\frac{{(2x-1)}^2}{9}}}+c$