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Popular >

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

Solution

\int \frac{e ^{x}}{e ^{2 x}} d x

Solution

- e^{- x} + C

Solution steps

\int \frac{e ^{x}}{e ^{2 x}} d x

Apply exponent rule:

\frac{a ^{n}}{a ^{m}} = a^{n - m}

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

= \int e^{- x} d x

Apply u-substitution

: \int - e^{u} d u

= \int - e^{u} d u

Take the constant out:

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

= - \int e^{u} d u

Use the common integral:

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

= - e^{u}

Substitute back

u = - x

= - e^{- x}

Add a constant to the solution

= - e^{- x} + C

Graph

Popular Examples

integral of (16x)/(8x^2+2)

\int \frac{16 x}{8 x ^{2} + 2} d x

integral of (2sin^2(x))/6

\int \frac{2 sin ^{2} ( x )}{6} d x

integral of x/(sqrt(-2x-x^2))

\int \frac{x}{- 2 x - x ^{2}} d x

integral of (-5)/(x^2sqrt(5+x^2))

\int \frac{- 5}{x ^{2} 5 + x ^{2}} d x

integral of (cos(pi/(x^4)))/(x^5)

\int \frac{cos ( \frac{π}{x ^{4}} )}{x ^{5}} d x

Frequently Asked Questions (FAQ)

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