What is the integral of 5sqrt(9+e^x) ?

The integral of 5sqrt(9+e^x) is 5(-3ln| 1/3 sqrt(e^x+9)+1|+3ln| 1/3 sqrt(e^x+9)-1|+2sqrt(e^x+9))+C

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integral of 5sqrt(9+e^x)

\int 5 9 + e^{x} d x

5 (- 3 ln \frac{1}{3} e^{x} + 9 + 1 + 3 ln \frac{1}{3} e^{x} + 9 - 1 + 2 e^{x} + 9) + C

Solution steps

\int 5 9 + e^{x} d x

Take the constant out:

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

= 5 \cdot \int 9 + e^{x} d x

Apply Integration By Parts

: x 9 + e^{x} - \int \frac{e ^{x} x}{2 9 + e ^{x}} d x

= 5 (x 9 + e^{x} - \int \frac{e ^{x} x}{2 9 + e ^{x}} d x)

\int \frac{e ^{x} x}{2 9 + e ^{x}} d x = x e^{x} + 9 + 6 (\frac{1}{2} ln \frac{1}{3} e^{x} + 9 + 1 - \frac{1}{2} ln \frac{1}{3} e^{x} + 9 - 1 - \frac{1}{3} e^{x} + 9)

= 5 (x 9 + e^{x} - (x e^{x} + 9 + 6 (\frac{1}{2} ln \frac{1}{3} e^{x} + 9 + 1 - \frac{1}{2} ln \frac{1}{3} e^{x} + 9 - 1 - \frac{1}{3} e^{x} + 9)))

Expand

x 9 + e^{x} - (x e^{x} + 9 + 6 (\frac{1}{2} ln \frac{1}{3} e^{x} + 9 + 1 - \frac{1}{2} ln \frac{1}{3} e^{x} + 9 - 1 - \frac{1}{3} e^{x} + 9)) : - 3 ln \frac{1}{3} e^{x} + 9 + 1 + 3 ln \frac{1}{3} e^{x} + 9 - 1 + 2 e^{x} + 9

= 5 (- 3 ln \frac{1}{3} e^{x} + 9 + 1 + 3 ln \frac{1}{3} e^{x} + 9 - 1 + 2 e^{x} + 9)

Add a constant to the solution

= 5 (- 3 ln \frac{1}{3} e^{x} + 9 + 1 + 3 ln \frac{1}{3} e^{x} + 9 - 1 + 2 e^{x} + 9) + C

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What is the integral of 5sqrt(9+e^x) ?
The integral of 5sqrt(9+e^x) is 5(-3ln| 1/3 sqrt(e^x+9)+1|+3ln| 1/3 sqrt(e^x+9)-1|+2sqrt(e^x+9))+C