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# Evaluate the following integrals:

Question:

Evaluate the following integrals: $\int\left(2 \mathrm{x}^{2}+3\right) \sqrt{\mathrm{x}+2} \mathrm{dx}$

Solution:

Let $\mathrm{I}=\int\left(2 \mathrm{x}^{2}+3\right) \sqrt{\mathrm{x}+2} \mathrm{~d} \mathrm{x}$

Substituting $x+2=t \Rightarrow d x=d t$

$\Rightarrow I=\int\left[2(t-2)^{2}+3\right] \sqrt{t} d t$

$\Rightarrow I=\int\left[2 t^{2}-8 t+8+3\right] \sqrt{t} d t$

$\Rightarrow I=\int\left[2 t^{\frac{5}{2}}-8 t^{\frac{3}{2}}+11 \frac{1}{2}\right] d t$

$\Rightarrow I=\frac{4}{7} t^{\frac{7}{2}}-\frac{16}{5} t^{\frac{5}{2}}+\frac{22}{3} t^{\frac{3}{2}}+c$

$\Rightarrow \mathrm{I}=\frac{4}{7}(\mathrm{x}+2)^{\frac{7}{2}}-\frac{16}{5}(\mathrm{x}+2)^{\frac{5}{2}}+\frac{22}{3}(\mathrm{x}+2)^{\frac{3}{2}}+\mathrm{c}$

$\therefore \int\left(2 x^{2}+3\right) \sqrt{x+2} d x=\frac{4}{7}(x+2)^{\frac{7}{2}}-\frac{16}{5}(x+2)^{\frac{5}{2}}+\frac{22}{3}(x+2)^{\frac{3}{2}}+c$