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

The integral from 0 to ln(a) of 6e^{3x} is 2(a^3-1)

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

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

2 (a^{3} - 1)

Solution steps

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

Apply u-substitution

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

= 6 \cdot \int_{0}^{3 l n (a)} 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 (a)} 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 (a)}

Simplify

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

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

Compute the boundaries:

a^{3} - 1

= 2 (a^{3} - 1)

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What is the integral from 0 to ln(a) of 6e^{3x} ?
The integral from 0 to ln(a) of 6e^{3x} is 2(a^3-1)