What is the integral of 3x^25^{4x} ?

The integral of 3x^25^{4x} is 3/(32ln^3(5))*625^x(8ln^2(5)x^2-4ln(5)x+1)+C

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integral of 3x^25^{4x}

\int 3 x^{2} 5^{4 x} d x

\frac{3}{32 ln ^{3} ( 5 )} \cdot 62 5^{x} (8 ln^{2} (5) x^{2} - 4 ln (5) x + 1) + C

Solution steps

\int 3 x^{2} \cdot 5^{4 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int x^{2} \cdot 5^{4 x} d x

Apply u-substitution

: \int \frac{5 ^{u} u ^{2}}{64} d u

= 3 \cdot \int \frac{5 ^{u} u ^{2}}{64} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \frac{1}{64} \cdot \int 5^{u} u^{2} d u

Apply Integration By Parts

: \frac{5 ^{u} u ^{2}}{ln ( 5 )} - \int \frac{2 \cdot 5 ^{u} u}{ln ( 5 )} d u

= 3 \cdot \frac{1}{64} (\frac{5 ^{u} u ^{2}}{ln ( 5 )} - \int \frac{2 \cdot 5 ^{u} u}{ln ( 5 )} d u)

\int \frac{2 \cdot 5 ^{u} u}{ln ( 5 )} d u = \frac{2}{ln ( 5 )} (\frac{5 ^{u} u}{ln ( 5 )} - \frac{5 ^{u}}{ln ^{2} ( 5 )})

= 3 \cdot \frac{1}{64} (\frac{5 ^{u} u ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{u} u}{ln ( 5 )} - \frac{5 ^{u}}{ln ^{2} ( 5 )}))

Substitute back

u = 4 x

= 3 \cdot \frac{1}{64} (\frac{5 ^{4 x} ( 4 x ) ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{4 x} \cdot 4 x}{ln ( 5 )} - \frac{5 ^{4 x}}{ln ^{2} ( 5 )}))

Simplify

3 \cdot \frac{1}{64} (\frac{5 ^{4 x} ( 4 x ) ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{4 x} \cdot 4 x}{ln ( 5 )} - \frac{5 ^{4 x}}{ln ^{2} ( 5 )})) : \frac{3}{32 ln ^{3} ( 5 )} \cdot 62 5^{x} (8 ln^{2} (5) x^{2} - 4 ln (5) x + 1)

= \frac{3}{32 ln ^{3} ( 5 )} \cdot 62 5^{x} (8 ln^{2} (5) x^{2} - 4 ln (5) x + 1)

Add a constant to the solution

= \frac{3}{32 ln ^{3} ( 5 )} \cdot 62 5^{x} (8 ln^{2} (5) x^{2} - 4 ln (5) x + 1) + C

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What is the integral of 3x^25^{4x} ?
The integral of 3x^25^{4x} is 3/(32ln^3(5))*625^x(8ln^2(5)x^2-4ln(5)x+1)+C