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Popular Calculus >

integral of x^45^x

Solution

\int x^{4} 5^{x} d x

Solution

\frac{5 ^{x} x ^{4}}{ln ( 5 )} - \frac{4}{ln ( 5 )} (\frac{5 ^{x} x ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{x} x}{ln ( 5 )} - \frac{5 ^{x}}{ln ^{2} ( 5 )}))) + C

Solution steps

\int x^{4} \cdot 5^{x} d x

Apply Integration By Parts

= \frac{5 ^{x} x ^{4}}{ln ( 5 )} - \int \frac{4 \cdot 5 ^{x} x ^{3}}{ln ( 5 )} d x

\int \frac{4 \cdot 5 ^{x} x ^{3}}{ln ( 5 )} d x = \frac{4}{ln ( 5 )} (\frac{5 ^{x} x ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{x} x}{ln ( 5 )} - \frac{5 ^{x}}{ln ^{2} ( 5 )})))

= \frac{5 ^{x} x ^{4}}{ln ( 5 )} - \frac{4}{ln ( 5 )} (\frac{5 ^{x} x ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{x} x}{ln ( 5 )} - \frac{5 ^{x}}{ln ^{2} ( 5 )})))

Add a constant to the solution

= \frac{5 ^{x} x ^{4}}{ln ( 5 )} - \frac{4}{ln ( 5 )} (\frac{5 ^{x} x ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{x} x}{ln ( 5 )} - \frac{5 ^{x}}{ln ^{2} ( 5 )}))) + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of x^45^x ?
The integral of x^45^x is