Evaluate the following integrals:

Question:

Evaluate $\int \mathrm{x} \sqrt{1+\mathrm{x}-\mathrm{x}^{2}} \mathrm{dx}$

Solution:

Make perfect square of quadratic equation

$1+x-x^{2}=\frac{5}{4}-\left(x^{2}-2\left(\frac{1}{2}\right)(x)+\left(\frac{1}{2}\right)^{2}\right)$

$=\left(\frac{\sqrt{5}}{2}\right)^{2}-\left(x-\frac{1}{2}\right)^{2}$

$y=\int x \sqrt{\left(\frac{\sqrt{5}}{2}\right)^{2}-\left(x-\frac{1}{2}\right)^{2}} d x$

Let, $x-\frac{1}{2}=t \Rightarrow x=t+\frac{1}{2}$

Differentiate both side with respect to $\mathrm{t}$

$\frac{d x}{d t}=1 \Rightarrow d x=d t$

$y=\int\left(t+\frac{1}{2}\right) \sqrt{\left(\frac{\sqrt{5}}{2}\right)^{2}-t^{2}} d t$

$y=\int t \sqrt{\left(\frac{\sqrt{5}}{2}\right)^{2}-t^{2}}+\frac{1}{2} \sqrt{\left(\frac{\sqrt{5}}{2}\right)^{2}-t^{2}} d t$

$A=\int t \sqrt{\left(\frac{\sqrt{5}}{2}\right)^{2}-t^{2}} d t$

Let, $t^{2}=z$

Differentiate both side with respect to $z$

$2 t \frac{d t}{d z}=1 \Rightarrow t d t=\frac{1}{2} d z$

$A=\frac{1}{2} \int \sqrt{\left(\frac{\sqrt{5}}{2}\right)^{2}-z} d z$

$A=\frac{1}{4} \int \sqrt{5-4 z} d z$

$A=\frac{-1}{24}(5-4 z)^{\frac{3}{2}}+c_{1}$

Put $z=t^{2}$ and $t=x-\frac{1}{2}$

$A=\frac{-1}{24}\left(5-4\left(x-\frac{1}{2}\right)^{2}\right)^{\frac{3}{2}}+c_{1}$

$A=\frac{-1}{3}\left(1+x-x^{2}\right)^{\frac{3}{2}}+c_{1}$

$B=\int \frac{1}{2} \sqrt{\left(\frac{\sqrt{5}}{2}\right)^{2}-t^{2}} d t$

$B=\frac{1}{2}\left(\frac{\left(\frac{\sqrt{5}}{2}\right)^{2}}{2} \sin ^{-1} \frac{t}{\left(\frac{\sqrt{5}}{2}\right)}+\frac{t}{2} \sqrt{\left(\frac{\sqrt{5}}{2}\right)^{2}-t^{2}}\right)+c_{2}$

$B=\frac{5}{16} \sin ^{-1}\left(\frac{2 t}{\sqrt{5}}\right)+\frac{t}{8} \sqrt{5-4 t^{2}}+c_{2}$

Put $t=x-\frac{1}{2}$

$B=\frac{5}{16} \sin ^{-1}\left(\frac{2 x-1}{\sqrt{5}}\right)+\frac{\left(x-\frac{1}{2}\right)}{8} \sqrt{5-4\left(x-\frac{1}{2}\right)^{2}}+c_{2}$

$B=\frac{5}{16} \sin ^{-1}\left(\frac{2 x-1}{\sqrt{5}}\right)+\frac{(2 x-1)}{8} \sqrt{1+x-x^{2}}+c_{2}$

The final answer is $\mathrm{y}=\mathrm{A}+\mathrm{B}$

$y=\frac{-1}{3}\left(1+x-x^{2}\right)^{\frac{3}{2}}+\frac{5}{16} \sin ^{-1}\left(\frac{2 x-1}{\sqrt{5}}\right)+\frac{(2 x-1)}{8} \sqrt{1+x-x^{2}}+c$

$y=\frac{1}{24}\left(8 x^{2}-2 x-11\right) \sqrt{1+x-x^{2}}+\frac{5}{16} \sin ^{-1}\left(\frac{2 x-1}{\sqrt{5}}\right)+c$