What is the integral of x(x^3+5x) ?

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

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integral of x(x^3+5x)

\int x (x^{3} + 5 x) d x

\frac{1}{2} x^{2} (x^{3} + 5 x) - \frac{1}{2} (\frac{3 x ^{5}}{5} + \frac{5 x ^{3}}{3}) + C

Solution steps

\int x (x^{3} + 5 x) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} (x^{3} + 5 x) - \int \frac{1}{2} x^{2} (3 x^{2} + 5) d x

= \frac{1}{2} x^{2} (x^{3} + 5 x) - \int \frac{1}{2} x^{2} (3 x^{2} + 5) d x

\int \frac{1}{2} x^{2} (3 x^{2} + 5) d x = \frac{1}{2} (\frac{3 x ^{5}}{5} + \frac{5 x ^{3}}{3})

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

Add a constant to the solution

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

\frac{d ^{2}}{d x ^{2}} (\frac{- x ^{2}}{x ^{2} - 2 x + 8})

\frac{d y}{d x} = \frac{e ^{x}}{3 + e ^{x}}

\int_{0}^{2} 1 + x^{3} d x

(x^{3} - 3 x) (2 x^{2} + 3 x + 5)

\int \frac{4}{( x - 3 ) ^{2}} d x

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