Mathematics

Integrate the following functions w.r.t $$X :(2x+1)\sqrt{x+2}$$


SOLUTION
Let $$I=\int (2x+1)\sqrt{x+2}dx$$
Put $$x+2=t$$
$$\therefore dx=dt$$
Alos, $$x=t-2$$
$$\therefore 2x+1=2(t-2)+1=2t-3$$
$$\therefore I=\displaystyle \int (2t-3)\sqrt{t}dt$$
$$=\displaystyle \int (2t^{\dfrac{3}{2}} - 3t^{\dfrac{1}{2}}) dt$$
$$=2\displaystyle \int t^{\dfrac{3}{2}}dt - 3 \int t^{\dfrac{1}{2}}dt$$
$$=2. \dfrac{t^\dfrac{5}{2}}{\left(\dfrac{5}{2}\right)} -3 \dfrac{t^{\dfrac{3}{2}}}{\left(\dfrac{3}{2}\right)}+c$$
$$=\dfrac{4}{5}(x+2)^{\dfrac{5}{2}}-2(x+2)^{\dfrac{3}{2}}+c$$
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Subjective Medium Published on 17th 09, 2020
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