What is the integral of x^2a^x ?

The integral of x^2a^x is (a^xx^2)/(ln(a))-2/(ln(a))((a^xx)/(ln(a))-(a^x)/(ln^2(a)))+C

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integral of x^2a^x

\int x^{2} a^{x} d x

\frac{a ^{x} x ^{2}}{ln ( a )} - \frac{2}{ln ( a )} (\frac{a ^{x} x}{ln ( a )} - \frac{a ^{x}}{ln ^{2} ( a )}) + C

Solution steps

\int x^{2} a^{x} d x

Apply Integration By Parts

: \frac{a ^{x} x ^{2}}{ln ( a )} - \int \frac{2 a ^{x} x}{ln ( a )} d x

= \frac{a ^{x} x ^{2}}{ln ( a )} - \int \frac{2 a ^{x} x}{ln ( a )} d x

\int \frac{2 a ^{x} x}{ln ( a )} d x = \frac{2}{ln ( a )} (\frac{a ^{x} x}{ln ( a )} - \frac{a ^{x}}{ln ^{2} ( a )})

= \frac{a ^{x} x ^{2}}{ln ( a )} - \frac{2}{ln ( a )} (\frac{a ^{x} x}{ln ( a )} - \frac{a ^{x}}{ln ^{2} ( a )})

Add a constant to the solution

= \frac{a ^{x} x ^{2}}{ln ( a )} - \frac{2}{ln ( a )} (\frac{a ^{x} x}{ln ( a )} - \frac{a ^{x}}{ln ^{2} ( a )}) + C

\int \frac{ln ( 2 x )}{x ln ( 4 x )} d x

\int \frac{ln ( ln ( 3 x ) )}{x ln ( 3 x )} d x

2 x^{2}

\int \frac{1}{1 - \frac{x ^{2}}{64}} d x

What is the integral of x^2a^x ?
The integral of x^2a^x is (a^xx^2)/(ln(a))-2/(ln(a))((a^xx)/(ln(a))-(a^x)/(ln^2(a)))+C