What is the integral of x*e^{xy} ?

The integral of x*e^{xy} is e^{xy}+C

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integral of x*e^{xy}

\int x \cdot e^{x y} d y

e^{x y} + C

Solution steps

\int x e^{x y} d y

Take the constant out:

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

= x \cdot \int e^{x y} d y

Apply u-substitution

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

= x \cdot \int \frac{e ^{u}}{x} d u

Take the constant out:

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

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

Use the common integral:

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

= x \frac{1}{x} e^{u}

Substitute back

u = x y

= x \frac{1}{x} e^{x y}

Simplify

x \frac{1}{x} e^{x y} : e^{x y}

= e^{x y}

Add a constant to the solution

= e^{x y} + C

\int 16 x e^{4 x} d x

x^{2} + 16

3 x^{3} - 12 x^{2} + 15 x

f (t) = t^{2}

\int 8 y^{- 3} d y