What is the integral of ((2e^x)/(e^{2-5x)})^2 ?

The integral of ((2e^x)/(e^{2-5x)})^2 is 1/3 e^{12x-4}+C

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integral of ((2e^x)/(e^{2-5x)})^2

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

\frac{1}{3} e^{12 x - 4} + C

Solution steps

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

Simplify

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

= \int \frac{2 e ^{x}}{e ^{2 - 5 x}} \cdot \frac{2 e ^{x}}{e ^{2 - 5 x}} d x

Take the constant out:

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

= 2 \cdot 2 \cdot \int \frac{e ^{x}}{e ^{2 - 5 x}} \cdot \frac{e ^{x}}{e ^{2 - 5 x}} d x

Simplify

= 4 \cdot \int \frac{e ^{x}}{e ^{2 - 5 x}} \cdot \frac{e ^{x}}{e ^{2 - 5 x}} d x

Simplify

\frac{e ^{x}}{e ^{2 - 5 x}} \cdot \frac{e ^{x}}{e ^{2 - 5 x}} : e^{12 x - 4}

= 4 \cdot \int e^{12 x - 4} d x

Apply u-substitution

= 4 \cdot \int e^{u} \frac{1}{12} d u

Take the constant out:

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

= 4 \cdot \frac{1}{12} \cdot \int e^{u} d u

Use the common integral:

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

= 4 \cdot \frac{1}{12} e^{u}

Substitute back

u = 12 x - 4

= 4 \cdot \frac{1}{12} e^{12 x - 4}

Simplify

4 \cdot \frac{1}{12} e^{12 x - 4} : \frac{1}{3} e^{12 x - 4}

= \frac{1}{3} e^{12 x - 4}

Add a constant to the solution

= \frac{1}{3} e^{12 x - 4} + C

What is the integral of ((2e^x)/(e^{2-5x)})^2 ?
The integral of ((2e^x)/(e^{2-5x)})^2 is 1/3 e^{12x-4}+C