Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int(3 x \sqrt{x}+4 \sqrt{x}+5) d x$

Solution:

$\int(3 x \sqrt{x}+4 \sqrt{x}+5) d x$

By Splitting, we get,

$\Rightarrow \int((3 \mathrm{x} \sqrt{\mathrm{x}}) \mathrm{dx}+(4 \sqrt{\mathrm{x}}) \mathrm{dx}+5 \mathrm{dx})$

$\Rightarrow \int 3 \mathrm{x} \sqrt{\mathrm{x}} \mathrm{dx}+\int 4 \sqrt{\mathrm{x}} \mathrm{dx}+\int 5 \mathrm{dx}$

$\Rightarrow \int 3 \mathrm{x}^{\frac{3}{2}} \mathrm{dx}+\int 4 \mathrm{x}^{\left(\frac{1}{2}\right)} \mathrm{d}+\int 5 \mathrm{dx}$

By using the formula, $\int x^{n} d x=\frac{x^{n+1}}{n+1}$

$\Rightarrow \frac{3 x^{\frac{3}{2}}+1}{\frac{2}{2}+1}+\frac{4 x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\int 5 d x$

$\int \mathrm{kdx}=\mathrm{kx}+\mathrm{c}$

$\Rightarrow \frac{3 x^{\frac{5}{2}}}{5 / 2}+\frac{4 x^{\frac{2}{2}}}{5 / 2}+5 x+c$

$\Rightarrow \frac{6}{5} x^{\frac{5}{2}}+\frac{4}{5} x^{3 / 2}+5 x+c$

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