What is the integral from 1 to 5 of 5/2 e^{0.4x} ?

The integral from 1 to 5 of 5/2 e^{0.4x} is 36.85769…

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integral from 1 to 5 of 5/2 e^{0.4x}

\int_{1}^{5} \frac{5}{2} e^{0.4 x} d x

36.85769 \dots

Solution steps

\int_{1}^{5} \frac{5}{2} e^{0.4 x} d x

Take the constant out:

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

= \frac{5}{2} \cdot \int_{1}^{5} e^{0.4 x} d x

Apply u-substitution

: \int_{0.4}^{2} e^{u} \frac{1}{0.4} d u

= \frac{5}{2} \cdot \int_{0.4}^{2} e^{u} \frac{1}{0.4} d u

Take the constant out:

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

= \frac{5}{2} \cdot \frac{1}{0.4} \cdot \int_{0.4}^{2} e^{u} d u

Simplify

= \frac{5}{2} \cdot 2.5 \cdot \int_{0.4}^{2} e^{u} d u

Use the common integral:

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

= \frac{5}{2} \cdot 2.5 [e^{u}]_{0.4}^{2}

Simplify

\frac{5}{2} \cdot 2.5 [e^{u}]_{0.4}^{2} : 6.25 [e^{u}]_{0.4}^{2}

= 6.25 [e^{u}]_{0.4}^{2}

Compute the boundaries:

5.89723 \dots

= 6.25 \cdot 5.89723 \dots

Simplify

= 36.85769 \dots

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What is the integral from 1 to 5 of 5/2 e^{0.4x} ?
The integral from 1 to 5 of 5/2 e^{0.4x} is 36.85769…