What is the integral of (e^{5x}+5e^x+8)/(e^x) ?

The integral of (e^{5x}+5e^x+8)/(e^x) is -(e^{5x}+5e^x+8)/(e^x)+5(1/4 e^{4x}+x)+C

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integral of (e^{5x}+5e^x+8)/(e^x)

\int \frac{e ^{5 x} + 5 e ^{x} + 8}{e ^{x}} d x

- \frac{e ^{5 x} + 5 e ^{x} + 8}{e ^{x}} + 5 (\frac{1}{4} e^{4 x} + x) + C

Solution steps

\int \frac{e ^{5 x} + 5 e ^{x} + 8}{e ^{x}} d x

Apply Integration By Parts

: - \frac{e ^{5 x} + 5 e ^{x} + 8}{e ^{x}} - \int - 5 (e^{4 x} + 1) d x

= - \frac{e ^{5 x} + 5 e ^{x} + 8}{e ^{x}} - \int - 5 (e^{4 x} + 1) d x

\int - 5 (e^{4 x} + 1) d x = - 5 (\frac{1}{4} e^{4 x} + x)

= - \frac{e ^{5 x} + 5 e ^{x} + 8}{e ^{x}} - (- 5 (\frac{1}{4} e^{4 x} + x))

Simplify

= - \frac{e ^{5 x} + 5 e ^{x} + 8}{e ^{x}} + 5 (\frac{1}{4} e^{4 x} + x)

Add a constant to the solution

= - \frac{e ^{5 x} + 5 e ^{x} + 8}{e ^{x}} + 5 (\frac{1}{4} e^{4 x} + x) + C

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What is the integral of (e^{5x}+5e^x+8)/(e^x) ?
The integral of (e^{5x}+5e^x+8)/(e^x) is -(e^{5x}+5e^x+8)/(e^x)+5(1/4 e^{4x}+x)+C