Mathematics

Find integrals of given function; $$\displaystyle \int{{e^{\sqrt x }}} \sqrt x dx = $$


SOLUTION
$$\displaystyle =\int e^{\sqrt{x}}\sqrt{x}dx$$

$$\sqrt{x}=t$$    $$x=t^{2}$$

$$\dfrac{dx}{2\sqrt{x}}=dt$$      $$\dfrac{\sqrt{x}dx}{2x}=dt\sqrt{x}=2xdt=2t^{2}dt$$
$$=\displaystyle \int 2t^{2}e^{t}dt$$

$$=\displaystyle 2\left[t^{2}e^{t}-\int 2te^{t}dt\right]$$
$$=\displaystyle 2\left[t^{2}e^{t}-2te^{t}+2\int e^{t}dt\right]$$

$$=2\left[t^{2}e^{t}-2te^{t}+2e^{t}\right]+C$$

put $$t=\sqrt{x}$$

$$=2xe^{\sqrt{x}}-4\sqrt{x}e^{\sqrt{x}}+4e^{\sqrt{x}}+C$$.
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Subjective Medium Published on 17th 09, 2020
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