What is the integral of x^3((2x^2+1)/x) ?

The integral of x^3((2x^2+1)/x) is -2x^5+(12x^5)/5+(x^3)/3+C

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integral of x^3((2x^2+1)/x)

\int x^{3} (\frac{2 x ^{2} + 1}{x}) d x

- 2 x^{5} + \frac{12 x ^{5}}{5} + \frac{x ^{3}}{3} + C

Solution steps

\int x^{3} \frac{2 x ^{2} + 1}{x} d x

\frac{2 x ^{2} + 1}{x} = (\frac{2 x ^{2}}{x} + \frac{1}{x})

= \int x^{3} (\frac{2 x ^{2}}{x} + \frac{1}{x}) d x

Cancel the common factor:

x

= \int x^{3} (2 x + \frac{1}{x}) d x

Apply Integration By Parts

: x^{4} (2 x + \frac{1}{x}) - \int x (8 x^{3} + 2 x) d x

= x^{4} (2 x + \frac{1}{x}) - \int x (8 x^{3} + 2 x) d x

\int x (8 x^{3} + 2 x) d x = 4 x^{5} + x^{3} - \frac{12 x ^{5}}{5} - \frac{x ^{3}}{3}

= x^{4} (2 x + \frac{1}{x}) - (4 x^{5} + x^{3} - \frac{12 x ^{5}}{5} - \frac{x ^{3}}{3})

Simplify

x^{4} (2 x + \frac{1}{x}) - (4 x^{5} + x^{3} - \frac{12 x ^{5}}{5} - \frac{x ^{3}}{3}) : - 2 x^{5} + \frac{12 x ^{5}}{5} + \frac{x ^{3}}{3}

= - 2 x^{5} + \frac{12 x ^{5}}{5} + \frac{x ^{3}}{3}

Add a constant to the solution

= - 2 x^{5} + \frac{12 x ^{5}}{5} + \frac{x ^{3}}{3} + C

\frac{x}{x + 3}

\int_{0}^{2} 5^{- x} d x

x y^{^{'}} + 15 y = 0

\frac{d}{d x} (sin (3 x^{3}))

\int \frac{2 x - 13}{x ^{2} - 13 x + 50} d x

What is the integral of x^3((2x^2+1)/x) ?
The integral of x^3((2x^2+1)/x) is -2x^5+(12x^5)/5+(x^3)/3+C