What is the integral of (e^{3x}+3e^x+3)/(e^x) ?

The integral of (e^{3x}+3e^x+3)/(e^x) is -(e^{3x}+3e^x+3)/(e^x)+3(1/2 e^{2x}+x)+C

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integral of (e^{3x}+3e^x+3)/(e^x)

\int \frac{e ^{3 x} + 3 e ^{x} + 3}{e ^{x}} d x

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

Solution steps

\int \frac{e ^{3 x} + 3 e ^{x} + 3}{e ^{x}} d x

Apply Integration By Parts

: - \frac{e ^{3 x} + 3 e ^{x} + 3}{e ^{x}} - \int - 3 (e^{2 x} + 1) d x

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (e^{3x}+3e^x+3)/(e^x) ?
The integral of (e^{3x}+3e^x+3)/(e^x) is -(e^{3x}+3e^x+3)/(e^x)+3(1/2 e^{2x}+x)+C