What is the integral of 2xe^{2y} ?

The integral of 2xe^{2y} is xe^{2y}+C

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integral of 2xe^{2y}

\int 2 x e^{2 y} d y

x e^{2 y} + C

Solution steps

\int 2 x e^{2 y} d y

Take the constant out:

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

= 2 x \cdot \int e^{2 y} d y

Apply u-substitution

: \int e^{u} \frac{1}{2} d u

= 2 x \cdot \int e^{u} \frac{1}{2} d u

Take the constant out:

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

= 2 x \frac{1}{2} \cdot \int e^{u} d u

Use the common integral:

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

= 2 x \frac{1}{2} e^{u}

Substitute back

u = 2 y

= 2 x \frac{1}{2} e^{2 y}

Simplify

2 x \frac{1}{2} e^{2 y} : x e^{2 y}

= x e^{2 y}

Add a constant to the solution

= x e^{2 y} + C

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