What is the integral of 1/2 e^{sqrt(5x+6)} ?

The integral of 1/2 e^{sqrt(5x+6)} is 1/5 (e^{sqrt(5x+6)}sqrt(5x+6)-e^{sqrt(5x+6)})+C

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integral of 1/2 e^{sqrt(5x+6)}

\int \frac{1}{2} e^{5 x + 6} d x

\frac{1}{5} (e^{5 x + 6} 5 x + 6 - e^{5 x + 6}) + C

Solution steps

\int \frac{1}{2} e^{5 x + 6} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int e^{5 x + 6} d x

Apply u-substitution

: \int \frac{2 e ^{u} u}{5} d u

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

Take the constant out:

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

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

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= \frac{1}{2} \cdot \frac{2}{5} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{2} \cdot \frac{2}{5} (e^{u} u - e^{u})

Substitute back

u = 5 x + 6

= \frac{1}{2} \cdot \frac{2}{5} (e^{5 x + 6} 5 x + 6 - e^{5 x + 6})

Simplify

\frac{1}{2} \cdot \frac{2}{5} (e^{5 x + 6} 5 x + 6 - e^{5 x + 6}) : \frac{1}{5} (e^{5 x + 6} 5 x + 6 - e^{5 x + 6})

= \frac{1}{5} (e^{5 x + 6} 5 x + 6 - e^{5 x + 6})

Add a constant to the solution

= \frac{1}{5} (e^{5 x + 6} 5 x + 6 - e^{5 x + 6}) + C

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What is the integral of 1/2 e^{sqrt(5x+6)} ?
The integral of 1/2 e^{sqrt(5x+6)} is 1/5 (e^{sqrt(5x+6)}sqrt(5x+6)-e^{sqrt(5x+6)})+C