What is the integral of (x^3-7x)*e^{5x} ?

The integral of (x^3-7x)*e^{5x} is 1/5 e^{5x}x^3-3/25 e^{5x}x^2-169/125 e^{5x}x+169/625 e^{5x}+C

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integral of (x^3-7x)*e^{5x}

\int (x^{3} - 7 x) \cdot e^{5 x} d x

\frac{1}{5} e^{5 x} x^{3} - \frac{3}{25} e^{5 x} x^{2} - \frac{169}{125} e^{5 x} x + \frac{169}{625} e^{5 x} + C

Solution steps

\int (x^{3} - 7 x) e^{5 x} d x

Expand

(x^{3} - 7 x) e^{5 x} : e^{5 x} x^{3} - 7 e^{5 x} x

= \int e^{5 x} x^{3} - 7 e^{5 x} x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e^{5 x} x^{3} d x - \int 7 e^{5 x} x d x

\int e^{5 x} x^{3} d x = \frac{1}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x})))

\int 7 e^{5 x} x d x = \frac{7}{25} (5 e^{5 x} x - e^{5 x})

= \frac{1}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x}))) - \frac{7}{25} (5 e^{5 x} x - e^{5 x})

Simplify

\frac{1}{625} (125 e^{5 x} x^{3} - 3 (25 e^{5 x} x^{2} - 2 (5 e^{5 x} x - e^{5 x}))) - \frac{7}{25} (5 e^{5 x} x - e^{5 x}) : \frac{1}{5} e^{5 x} x^{3} - \frac{3}{25} e^{5 x} x^{2} - \frac{169}{125} e^{5 x} x + \frac{169}{625} e^{5 x}

= \frac{1}{5} e^{5 x} x^{3} - \frac{3}{25} e^{5 x} x^{2} - \frac{169}{125} e^{5 x} x + \frac{169}{625} e^{5 x}

Add a constant to the solution

= \frac{1}{5} e^{5 x} x^{3} - \frac{3}{25} e^{5 x} x^{2} - \frac{169}{125} e^{5 x} x + \frac{169}{625} e^{5 x} + C

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What is the integral of (x^3-7x)*e^{5x} ?
The integral of (x^3-7x)*e^{5x} is 1/5 e^{5x}x^3-3/25 e^{5x}x^2-169/125 e^{5x}x+169/625 e^{5x}+C