Find the integral: x^ 3 +3 x+4 x d x - Vidyadip

Question

Find the integral: $\int \frac{x^{3}+3 x+4}{\sqrt{x}} d x$

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Answer

Let I =$\int \frac{x^{3}+3 x+4}{\sqrt{x}} d x$
Separating the terms we get,
I =$\int\left(x^{\frac{5}{2}}+3 x^{\frac{1}{2}}+4 x^{-\frac{1}{2}}\right) d x$
Applying the formula,
$\int x^{n} d x=\frac{x^{n+1}}{n+1}+c$
I = $\frac{x^{\frac{7}{2}}}{\frac{7}{2}}+\frac{3\left(x^{\frac{3}{2}}\right)}{\frac{3}{2}}+\frac{4\left(x^{\frac{1}{2}}\right)}{\frac{1}{2}}+C$
= $\frac{2}{7} x^{\frac{7}{2}}+2 x^{\frac{3}{2}}+8 x^{\frac{1}{2}}+C$
= $\frac{2}{7} x^{\frac{7}{2}}+2 x^{\frac{3}{2}}+8 \sqrt{x}+C$

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