What is the integral of x^7e^x ?

The integral of x^7e^x is e^xx^7-7(e^xx^6-6(e^xx^5-5(e^xx^4-4(e^xx^3-3(e^xx^2-2(e^xx-e^x))))))+C

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integral of x^7e^x

\int x^{7} e^{x} d x

e^{x} x^{7} - 7 (e^{x} x^{6} - 6 (e^{x} x^{5} - 5 (e^{x} x^{4} - 4 (e^{x} x^{3} - 3 (e^{x} x^{2} - 2 (e^{x} x - e^{x})))))) + C

Solution steps

\int x^{7} e^{x} d x

Apply u-substitution

= \int ln^{7} (u) d u

Apply Integration By Parts

= u ln^{7} (u) - \int 7 ln^{6} (u) d u

\int 7 ln^{6} (u) d u = 7 (u ln^{6} (u) - 6 (u ln^{5} (u) - 5 (u ln^{4} (u) - 4 (u ln^{3} (u) - 3 (u ln^{2} (u) - 2 (u ln (u) - u))))))

= u ln^{7} (u) - 7 (u ln^{6} (u) - 6 (u ln^{5} (u) - 5 (u ln^{4} (u) - 4 (u ln^{3} (u) - 3 (u ln^{2} (u) - 2 (u ln (u) - u))))))

Substitute back

u = e^{x}

= e^{x} ln^{7} (e^{x}) - 7 (e^{x} ln^{6} (e^{x}) - 6 (e^{x} ln^{5} (e^{x}) - 5 (e^{x} ln^{4} (e^{x}) - 4 (e^{x} ln^{3} (e^{x}) - 3 (e^{x} ln^{2} (e^{x}) - 2 (e^{x} ln (e^{x}) - e^{x}))))))

Simplify

e^{x} ln^{7} (e^{x}) - 7 (e^{x} ln^{6} (e^{x}) - 6 (e^{x} ln^{5} (e^{x}) - 5 (e^{x} ln^{4} (e^{x}) - 4 (e^{x} ln^{3} (e^{x}) - 3 (e^{x} ln^{2} (e^{x}) - 2 (e^{x} ln (e^{x}) - e^{x})))))) : e^{x} x^{7} - 7 (e^{x} x^{6} - 6 (e^{x} x^{5} - 5 (e^{x} x^{4} - 4 (e^{x} x^{3} - 3 (e^{x} x^{2} - 2 (e^{x} x - e^{x}))))))

= e^{x} x^{7} - 7 (e^{x} x^{6} - 6 (e^{x} x^{5} - 5 (e^{x} x^{4} - 4 (e^{x} x^{3} - 3 (e^{x} x^{2} - 2 (e^{x} x - e^{x}))))))

Add a constant to the solution

= e^{x} x^{7} - 7 (e^{x} x^{6} - 6 (e^{x} x^{5} - 5 (e^{x} x^{4} - 4 (e^{x} x^{3} - 3 (e^{x} x^{2} - 2 (e^{x} x - e^{x})))))) + C

What is the integral of x^7e^x ?
The integral of x^7e^x is e^xx^7-7(e^xx^6-6(e^xx^5-5(e^xx^4-4(e^xx^3-3(e^xx^2-2(e^xx-e^x))))))+C