What is the integral of x^6e^{2x} ?

The integral of x^6e^{2x} is 1/8 e^{2x}(4x^6-3(4x^5-5(2x^4-4x^3+6x^2-6x+3)))+C

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integral of x^6e^{2x}

\int x^{6} e^{2 x} d x

\frac{1}{8} e^{2 x} (4 x^{6} - 3 (4 x^{5} - 5 (2 x^{4} - 4 x^{3} + 6 x^{2} - 6 x + 3))) + C

Solution steps

\int x^{6} e^{2 x} d x

Apply Integration By Parts

= \frac{1}{7} e^{2 x} x^{7} - \int \frac{2}{7} e^{2 x} x^{7} d x

\int \frac{2}{7} e^{2 x} x^{7} d x = \frac{1}{896} (128 e^{2 x} x^{7} - 7 (64 e^{2 x} x^{6} - 6 (32 e^{2 x} x^{5} - 5 (16 e^{2 x} x^{4} - 4 (8 e^{2 x} x^{3} - 3 (4 e^{2 x} x^{2} - 2 (2 e^{2 x} x - e^{2 x})))))))

= \frac{1}{7} e^{2 x} x^{7} - \frac{1}{896} (128 e^{2 x} x^{7} - 7 (64 e^{2 x} x^{6} - 6 (32 e^{2 x} x^{5} - 5 (16 e^{2 x} x^{4} - 4 (8 e^{2 x} x^{3} - 3 (4 e^{2 x} x^{2} - 2 (2 e^{2 x} x - e^{2 x})))))))

Simplify

\frac{1}{7} e^{2 x} x^{7} - \frac{1}{896} (128 e^{2 x} x^{7} - 7 (64 e^{2 x} x^{6} - 6 (32 e^{2 x} x^{5} - 5 (16 e^{2 x} x^{4} - 4 (8 e^{2 x} x^{3} - 3 (4 e^{2 x} x^{2} - 2 (2 e^{2 x} x - e^{2 x}))))))) : \frac{1}{8} e^{2 x} (4 x^{6} - 3 (4 x^{5} - 5 (2 x^{4} - 4 x^{3} + 6 x^{2} - 6 x + 3)))

= \frac{1}{8} e^{2 x} (4 x^{6} - 3 (4 x^{5} - 5 (2 x^{4} - 4 x^{3} + 6 x^{2} - 6 x + 3)))

Add a constant to the solution

= \frac{1}{8} e^{2 x} (4 x^{6} - 3 (4 x^{5} - 5 (2 x^{4} - 4 x^{3} + 6 x^{2} - 6 x + 3))) + C

What is the integral of x^6e^{2x} ?
The integral of x^6e^{2x} is 1/8 e^{2x}(4x^6-3(4x^5-5(2x^4-4x^3+6x^2-6x+3)))+C