What is the integral from 0 to ln(3) of (3e^{2x}) ?

The integral from 0 to ln(3) of (3e^{2x}) is 12

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integral from 0 to ln(3) of (3e^{2x})

\int_{0}^{l n (3)} (3 e^{2 x}) d x

12

Solution steps

\int_{0}^{l n (3)} 3 e^{2 x} d x

Take the constant out:

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

= 3 \cdot \int_{0}^{l n (3)} e^{2 x} d x

Apply u-substitution

: \int_{0}^{2 l n (3)} e^{u} \frac{1}{2} d u

= 3 \cdot \int_{0}^{2 l n (3)} 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

= 3 \cdot \frac{1}{2} \cdot \int_{0}^{2 l n (3)} e^{u} d u

Use the common integral:

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

= 3 \cdot \frac{1}{2} [e^{u}]_{0}^{2 l n (3)}

Simplify

3 \cdot \frac{1}{2} [e^{u}]_{0}^{2 l n (3)} : \frac{3}{2} [e^{u}]_{0}^{2 l n (3)}

= \frac{3}{2} [e^{u}]_{0}^{2 l n (3)}

Compute the boundaries:

8

= \frac{3}{2} \cdot 8

Simplify

= 12

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What is the integral from 0 to ln(3) of (3e^{2x}) ?
The integral from 0 to ln(3) of (3e^{2x}) is 12