What is the integral of e^{2x}y ?

The integral of e^{2x}y is y 1/2 e^{2x}+C

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integral of e^{2x}y

\int e^{2 x} y d x

y \frac{1}{2} e^{2 x} + C

Solution steps

\int e^{2 x} y d x

Take the constant out:

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

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

Apply u-substitution

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

= y \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

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

Use the common integral:

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

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

Substitute back

u = 2 x

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

Add a constant to the solution

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

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