Evaluate the following integrals:
Question:

Evaluate the following integrals:

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

Solution:

Given:

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

By multiplying $\sqrt{x}$ inside the bracket we get,

$\Rightarrow \int(3 \sqrt{x}-5 x \sqrt{x}) d x$

$\Rightarrow \int\left(3 x^{\frac{1}{2}}-5 x^{1} \times x^{\frac{1}{2}}\right) d x$

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

$\Rightarrow \int\left(3 x^{\frac{1}{2}}-5 x^{\frac{2}{2}}\right) d x$

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

By using the formula,

$\int x^{n} d x=\frac{x^{n+1}}{n+1}$

$\Rightarrow \frac{3 x^{\frac{1}{2}+1}}{\frac{1}{2}+1}-\frac{5 x^{\frac{3}{2}+1}}{\frac{3}{2}+1}+c$

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

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

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