What is the integral of 2e^{5x} ?

The integral of 2e^{5x} is 2/5 e^{5x}+C

integral of 2e^{5x}

\int 2 e^{5 x} d x

\frac{2}{5} e^{5 x} + C

Solution steps

\int 2 e^{5 x} d x

Take the constant out:

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

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

Apply u-substitution

= 2 \cdot \int e^{u} \frac{1}{5} d u

Take the constant out:

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

= 2 \cdot \frac{1}{5} \cdot \int e^{u} d u

Use the common integral:

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

= 2 \cdot \frac{1}{5} e^{u}

Substitute back

u = 5 x

= 2 \cdot \frac{1}{5} e^{5 x}

Simplify

2 \cdot \frac{1}{5} e^{5 x} : \frac{2}{5} e^{5 x}

= \frac{2}{5} e^{5 x}

Add a constant to the solution

= \frac{2}{5} e^{5 x} + C

n = 0 \sum \infty 2^{n}

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\frac{d y}{d x} = - \frac{1 + y ^{2}}{1 + x ^{2}}