What is the integral of xe^{5x+pi} ?

The integral of xe^{5x+pi} is 1/25 (5e^{5x+pi}x-e^{5x+pi})+C

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integral of xe^{5x+pi}

\int x e^{5 x + π} d x

\frac{1}{25} (5 e^{5 x + π} x - e^{5 x + π}) + C

Solution steps

\int x e^{5 x + π} d x

Apply u-substitution

: \int \frac{ln ( u ) - π}{25} d u

= \int \frac{ln ( u ) - π}{25} d u

Take the constant out:

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

= \frac{1}{25} \cdot \int ln (u) - π d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{25} (\int ln (u) d u - \int π d u)

\int ln (u) d u = u ln (u) - u

\int π d u = π u

= \frac{1}{25} (u ln (u) - u - π u)

Substitute back

u = e^{π + 5 x}

= \frac{1}{25} (e^{π + 5 x} ln (e^{π + 5 x}) - e^{π + 5 x} - π e^{π + 5 x})

Simplify

\frac{1}{25} (e^{π + 5 x} ln (e^{π + 5 x}) - e^{π + 5 x} - π e^{π + 5 x}) : \frac{1}{25} (5 e^{5 x + π} x - e^{5 x + π})

= \frac{1}{25} (5 e^{5 x + π} x - e^{5 x + π})

Add a constant to the solution

= \frac{1}{25} (5 e^{5 x + π} x - e^{5 x + π}) + C

x \to - 8 lim (- \frac{3 x}{x + 8})

3 x^{3} 25 - x^{2}

\int_{6}^{8} \frac{70}{( x - 6 ) ^{3}} d x

1 - 3 x^{2}

f^{^{'}} (x) = f (x) (1 - f (x))

What is the integral of xe^{5x+pi} ?
The integral of xe^{5x+pi} is 1/25 (5e^{5x+pi}x-e^{5x+pi})+C