What is the integral of 5x^3e^{5x-3} ?

The integral of 5x^3e^{5x-3} is e^{5x-3}x^3-3/5 e^{5x-3}x^2+6/125 e^{5x-3}(5x-1)+C

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integral of 5x^3e^{5x-3}

\int 5 x^{3} e^{5 x - 3} d x

e^{5 x - 3} x^{3} - \frac{3}{5} e^{5 x - 3} x^{2} + \frac{6}{125} e^{5 x - 3} (5 x - 1) + C

Solution steps

\int 5 x^{3} e^{5 x - 3} d x

Take the constant out:

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

= 5 \cdot \int x^{3} e^{5 x - 3} d x

x^{3} e^{5 x - 3} = e^{- 3} x^{3} e^{5 x}

= 5 \cdot \int e^{- 3} x^{3} e^{5 x} d x

Take the constant out:

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

= 5 e^{- 3} \cdot \int x^{3} e^{5 x} d x

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= 5 \cdot \frac{1}{e ^{3}} \cdot \int x^{3} e^{5 x} d x

Apply u-substitution

: \int \frac{e ^{u} u ^{3}}{625} d u

= 5 \cdot \frac{1}{e ^{3}} \cdot \int \frac{e ^{u} u ^{3}}{625} d u

Take the constant out:

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

= 5 \cdot \frac{1}{e ^{3}} \cdot \frac{1}{625} \cdot \int e^{u} u^{3} d u

Apply Integration By Parts

: u^{3} e^{u} - \int 3 u^{2} e^{u} d u

= 5 \cdot \frac{1}{e ^{3}} \cdot \frac{1}{625} (u^{3} e^{u} - \int 3 u^{2} e^{u} d u)

\int 3 u^{2} e^{u} d u = 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

= 5 \cdot \frac{1}{e ^{3}} \cdot \frac{1}{625} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Substitute back

u = 5 x

= 5 \cdot \frac{1}{e ^{3}} \cdot \frac{1}{625} ((5 x)^{3} e^{5 x} - 3 ((5 x)^{2} e^{5 x} - 2 (e^{5 x} \cdot 5 x - e^{5 x})))

Simplify

5 \cdot \frac{1}{e ^{3}} \cdot \frac{1}{625} ((5 x)^{3} e^{5 x} - 3 ((5 x)^{2} e^{5 x} - 2 (e^{5 x} \cdot 5 x - e^{5 x}))) : e^{5 x - 3} x^{3} - \frac{3}{5} e^{5 x - 3} x^{2} + \frac{6}{125} e^{5 x - 3} (5 x - 1)

= e^{5 x - 3} x^{3} - \frac{3}{5} e^{5 x - 3} x^{2} + \frac{6}{125} e^{5 x - 3} (5 x - 1)

Add a constant to the solution

= e^{5 x - 3} x^{3} - \frac{3}{5} e^{5 x - 3} x^{2} + \frac{6}{125} e^{5 x - 3} (5 x - 1) + C

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What is the integral of 5x^3e^{5x-3} ?
The integral of 5x^3e^{5x-3} is e^{5x-3}x^3-3/5 e^{5x-3}x^2+6/125 e^{5x-3}(5x-1)+C