What is the integral of (3e^{3x})/(x+10) ?

The integral of (3e^{3x})/(x+10) is 3/(e^{30)}\Ei(3x+30)+C

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integral of (3e^{3x})/(x+10)

\int \frac{3 e ^{3 x}}{x + 10} d x

\frac{3}{e ^{30}} Ei (3 x + 30) + C

Solution steps

\int \frac{3 e ^{3 x}}{x + 10} d x

Take the constant out:

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

= 3 \cdot \int \frac{e ^{3 x}}{x + 10} d x

Apply u-substitution

: \int \frac{e ^{u}}{u + 30} d u

= 3 \cdot \int \frac{e ^{u}}{u + 30} d u

Apply u-substitution

: \int \frac{e ^{v - 30}}{v} d v

= 3 \cdot \int \frac{e ^{v - 30}}{v} d v

\frac{e ^{v - 30}}{v} = e^{- 30} \frac{e ^{v}}{v}

= 3 \cdot \int e^{- 30} \frac{e ^{v}}{v} d v

Take the constant out:

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

= 3 e^{- 30} \cdot \int \frac{e ^{v}}{v} d v

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= 3 \cdot \frac{1}{e ^{30}} \cdot \int \frac{e ^{v}}{v} d v

Use the nonelementary integral:

\int \frac{e ^{x}}{x} d x = Ei (x)

= 3 \cdot \frac{1}{e ^{30}} Ei (v)

Substitute back

= 3 \cdot \frac{1}{e ^{30}} Ei (3 x + 30)

Simplify

3 \cdot \frac{1}{e ^{30}} Ei (3 x + 30) : \frac{3}{e ^{30}} Ei (3 x + 30)

= \frac{3}{e ^{30}} Ei (3 x + 30)

Add a constant to the solution

= \frac{3}{e ^{30}} Ei (3 x + 30) + C

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What is the integral of (3e^{3x})/(x+10) ?
The integral of (3e^{3x})/(x+10) is 3/(e^{30)}\Ei(3x+30)+C