What is the integral from 0 to 1 of e^{sqrt(3x+1)} ?

The integral from 0 to 1 of e^{sqrt(3x+1)} is (2e^2)/3

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integral from 0 to 1 of e^{sqrt(3x+1)}

\int_{0}^{1} e^{3 x + 1} d x

\frac{2 e ^{2}}{3}

Decimal

4.92603 \dots

Solution steps

\int_{0}^{1} e^{3 x + 1} d x

Apply u-substitution

: \int_{1}^{2} \frac{2 e ^{u} u}{3} d u

= \int_{1}^{2} \frac{2 e ^{u} u}{3} d u

Take the constant out:

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

= \frac{2}{3} \cdot \int_{1}^{2} e^{u} u d u

Apply Integration By Parts

: [e^{u} u - \int e^{u} d u]_{1}^{2}

= \frac{2}{3} [e^{u} u - \int e^{u} d u]_{1}^{2}

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

= \frac{2}{3} [e^{u} u - e^{u}]_{1}^{2}

Compute the boundaries:

e^{2}

= \frac{2}{3} e^{2}

Simplify

= \frac{2 e ^{2}}{3}

\int_{2}^{x} sin (x) e^{x} d x

\int_{0}^{2} \frac{1}{3} d x

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\int_{0}^{1} r 2 - r^{2} d r

\int_{0}^{1} z e^{- z} d z

What is the integral from 0 to 1 of e^{sqrt(3x+1)} ?
The integral from 0 to 1 of e^{sqrt(3x+1)} is (2e^2)/3