What is the integral of e^{6x}(x^2-11x+1) ?

The integral of e^{6x}(x^2-11x+1) is 1/6 e^{6x}x^2-17/9 e^{6x}x+13/27 e^{6x}+C

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integral of e^{6x}(x^2-11x+1)

\int e^{6 x} (x^{2} - 11 x + 1) d x

\frac{1}{6} e^{6 x} x^{2} - \frac{17}{9} e^{6 x} x + \frac{13}{27} e^{6 x} + C

Solution steps

\int e^{6 x} (x^{2} - 11 x + 1) d x

Expand

e^{6 x} (x^{2} - 11 x + 1) : e^{6 x} x^{2} - 11 e^{6 x} x + e^{6 x}

= \int e^{6 x} x^{2} - 11 e^{6 x} x + e^{6 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^{6 x} x^{2} d x - \int 11 e^{6 x} x d x + \int e^{6 x} d x

\int e^{6 x} x^{2} d x = \frac{1}{6} e^{6 x} x^{2} - \frac{1}{108} (6 e^{6 x} x - e^{6 x})

\int 11 e^{6 x} x d x = \frac{11}{36} (6 e^{6 x} x - e^{6 x})

\int e^{6 x} d x = \frac{1}{6} e^{6 x}

= \frac{1}{6} e^{6 x} x^{2} - \frac{1}{108} (6 e^{6 x} x - e^{6 x}) - \frac{11}{36} (6 e^{6 x} x - e^{6 x}) + \frac{1}{6} e^{6 x}

Simplify

\frac{1}{6} e^{6 x} x^{2} - \frac{1}{108} (6 e^{6 x} x - e^{6 x}) - \frac{11}{36} (6 e^{6 x} x - e^{6 x}) + \frac{1}{6} e^{6 x} : \frac{1}{6} e^{6 x} x^{2} - \frac{17}{9} e^{6 x} x + \frac{13}{27} e^{6 x}

= \frac{1}{6} e^{6 x} x^{2} - \frac{17}{9} e^{6 x} x + \frac{13}{27} e^{6 x}

Add a constant to the solution

= \frac{1}{6} e^{6 x} x^{2} - \frac{17}{9} e^{6 x} x + \frac{13}{27} e^{6 x} + C

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What is the integral of e^{6x}(x^2-11x+1) ?
The integral of e^{6x}(x^2-11x+1) is 1/6 e^{6x}x^2-17/9 e^{6x}x+13/27 e^{6x}+C