What is the integral of 1/4 e^{2x} ?

The integral of 1/4 e^{2x} is 1/8 e^{2x}+C

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integral of 1/4 e^{2x}

\int \frac{1}{4} e^{2 x} d x

\frac{1}{8} e^{2 x} + C

Solution steps

\int \frac{1}{4} e^{2 x} d x

Take the constant out:

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

= \frac{1}{4} \cdot \int e^{2 x} d x

Apply u-substitution

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

= \frac{1}{4} \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

= \frac{1}{4} \cdot \frac{1}{2} \cdot \int e^{u} d u

Use the common integral:

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

= \frac{1}{4} \cdot \frac{1}{2} e^{u}

Substitute back

u = 2 x

= \frac{1}{4} \cdot \frac{1}{2} e^{2 x}

Simplify

\frac{1}{4} \cdot \frac{1}{2} e^{2 x} : \frac{1}{8} e^{2 x}

= \frac{1}{8} e^{2 x}

Add a constant to the solution

= \frac{1}{8} e^{2 x} + C

\frac{d}{d x} (\frac{1}{2 x} + 2 x)

\int_{4}^{9} \frac{1}{7 - x} d x

\frac{d y}{d x} = e^{6 x} + 12 y

(\frac{1}{2 x + 3})^{3}

(- 1.2) (5.5)