Mathematics

Evaluate the following integral:
$$\int { \left( \sqrt { x } \left( a{ x }^{ 2 }+bx+c \right)  \right)  } dx$$


SOLUTION
Given $$\displaystyle\int{\left(\sqrt{x}\left(a{x}^{2}+bx+c\right)\right)dx}$$

$$=\displaystyle\int{\left(a{x}^{\frac{1}{2}+2}+b{x}^{1+\frac{1}{2}}+c{x}^{\frac{1}{2}}\right)dx}$$

$$=\displaystyle\int{\left(a{x}^{\frac{1+4}{2}}+b{x}^{\frac{2+1}{2}}+c{x}^{\frac{1}{2}}\right)dx}$$

$$=\displaystyle\int{\left(a{x}^{\frac{5}{2}}+b{x}^{\frac{3}{2}}+c{x}^{\frac{1}{2}}\right)dx}$$

$$=a\displaystyle\int{{x}^{\frac{5}{2}}dx}+b\displaystyle\int{{x}^{\frac{3}{2}}dx}+c\displaystyle\int{{x}^{\frac{1}{2}}dx}$$

We know that $$\displaystyle\int{{x}^{n}dx}=\dfrac{{x}^{n+1}}{n+1}+c$$

$$=\dfrac{a{x}^{\frac{5}{2}+1}}{\dfrac{5}{2}+1}+\dfrac{b{x}^{\frac{3}{2}+1}}{\dfrac{3}{2}+1}+\dfrac{c{x}^{\frac{1}{2}+1}}{\dfrac{1}{2}+1}+C$$ where $$C$$ is the constant of integration.

$$=\dfrac{a{x}^{\frac{5+2}{2}}}{\dfrac{5+2}{2}}+\dfrac{b{x}^{\frac{3+2}{2}}}{\dfrac{3+2}{2}}+\dfrac{c{x}^{\frac{1+2}{2}}}{\dfrac{1+2}{2}}+C$$

$$=\dfrac{a{x}^{\frac{7}{2}}}{\dfrac{7}{2}}+\dfrac{b{x}^{\frac{5}{2}}}{\dfrac{5}{2}}+\dfrac{c{x}^{\frac{3}{2}}}{\dfrac{3}{2}}+C$$

$$=\dfrac{2a{x}^{\frac{7}{2}}}{7}+\dfrac{2b{x}^{\frac{5}{2}}}{5}+\dfrac{2c{x}^{\frac{3}{2}}}{3}+C$$
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Subjective Medium Published on 17th 09, 2020
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