What is the integral from 0 to 2 of 2e^{2x} ?

The integral from 0 to 2 of 2e^{2x} is e^4-1

integral from 0 to 2 of 2e^{2x}

\int_{0}^{2} 2 e^{2 x} d x

e^{4} - 1

Decimal

53.59815 \dots

Solution steps

\int_{0}^{2} 2 e^{2 x} d x

Take the constant out:

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

= 2 \cdot \int_{0}^{2} e^{2 x} d x

Apply u-substitution

= 2 \cdot \int_{0}^{4} 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

= 2 \cdot \frac{1}{2} \cdot \int_{0}^{4} e^{u} d u

Use the common integral:

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

= 2 \cdot \frac{1}{2} [e^{u}]_{0}^{4}

Simplify

2 \cdot \frac{1}{2} [e^{u}]_{0}^{4} : [e^{u}]_{0}^{4}

= [e^{u}]_{0}^{4}

Compute the boundaries:

e^{4} - 1

= e^{4} - 1

\int_{1}^{2} 1 + \frac{9}{4} x d x

\int_{0}^{7} x^{3} - 9 x^{2} + 20 x d x

\int_{2}^{3} x^{2} - 3 x + 2 d x

\int_{0}^{5} 15 x x + 4 d x

\int_{0}^{310} x^{2} d x

What is the integral from 0 to 2 of 2e^{2x} ?
The integral from 0 to 2 of 2e^{2x} is e^4-1