What is the integral of xe^{sqrt(11)x+sqrt(3)} ?

The integral of xe^{sqrt(11)x+sqrt(3)} is (e^{sqrt(3)})/(11)(sqrt(11)e^{sqrt(11)x}x-e^{sqrt(11)x})+C

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integral of xe^{sqrt(11)x+sqrt(3)}

\int x e^{11 x + 3} d x

\frac{e ^{3}}{11} (11 e^{11 x} x - e^{11 x}) + C

Solution steps

\int x e^{11 x + 3} d x

x e^{11 x + 3} = e^{3} x e^{11 x}

= \int e^{3} x e^{11 x} d x

Take the constant out:

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

= e^{3} \cdot \int x e^{11 x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{11} d u

= e^{3} \cdot \int \frac{e ^{u} u}{11} d u

Take the constant out:

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

= e^{3} \frac{1}{11} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= e^{3} \frac{1}{11} (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

= e^{3} \frac{1}{11} (e^{u} u - e^{u})

Substitute back

u = 11 x

= e^{3} \frac{1}{11} (e^{11 x} 11 x - e^{11 x})

Simplify

e^{3} \frac{1}{11} (e^{11 x} 11 x - e^{11 x}) : \frac{e ^{3}}{11} (11 e^{11 x} x - e^{11 x})

= \frac{e ^{3}}{11} (11 e^{11 x} x - e^{11 x})

Add a constant to the solution

= \frac{e ^{3}}{11} (11 e^{11 x} x - e^{11 x}) + C

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What is the integral of xe^{sqrt(11)x+sqrt(3)} ?
The integral of xe^{sqrt(11)x+sqrt(3)} is (e^{sqrt(3)})/(11)(sqrt(11)e^{sqrt(11)x}x-e^{sqrt(11)x})+C