Find the integral: x^ 3 +5 x^ 2 -4 x^ 2 d x - Vidyadip

Question

Find the integral: $\int \frac{x^{3}+5 x^{2}-4}{x^{2}} d x$

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Answer

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

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