What is the integral from 0 to ln(2) of 6e^{3x} ?

The integral from 0 to ln(2) of 6e^{3x} is 14

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integral from 0 to ln(2) of 6e^{3x}

\int_{0}^{l n (2)} 6 e^{3 x} d x

14

Solution steps

\int_{0}^{l n (2)} 6 e^{3 x} d x

Take the constant out:

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

= 6 \cdot \int_{0}^{l n (2)} e^{3 x} d x

Apply u-substitution

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

= 6 \cdot \int_{0}^{3 l n (2)} e^{u} \frac{1}{3} d u

Take the constant out:

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

= 6 \cdot \frac{1}{3} \cdot \int_{0}^{3 l n (2)} e^{u} d u

Use the common integral:

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

= 6 \cdot \frac{1}{3} [e^{u}]_{0}^{3 l n (2)}

Simplify

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

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

Compute the boundaries:

7

= 2 \cdot 7

Simplify

= 14

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What is the integral from 0 to ln(2) of 6e^{3x} ?
The integral from 0 to ln(2) of 6e^{3x} is 14