What is the integral of 2/(e^t) ?

The integral of 2/(e^t) is -2/(e^t)+C

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integral of 2/(e^t)

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

- \frac{2}{e ^{t}} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{u ^{2}} d u

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

Apply the Power Rule

: - \frac{1}{u}

= 2 (- \frac{1}{u})

Substitute back

u = e^{t}

= 2 (- \frac{1}{e ^{t}})

Simplify

2 (- \frac{1}{e ^{t}}) : - \frac{2}{e ^{t}}

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

Add a constant to the solution

= - \frac{2}{e ^{t}} + C

\frac{\partial}{\partial y} (\frac{1}{x ^{2} + y ^{2} - 1})

\frac{d y}{d x} = \frac{2 x y}{( x ^{2} - y ^{2} )}

\frac{d y}{d x} - 5 y = - \frac{5}{2} x y^{3}

\int \frac{x}{x ^{2} + 6} d x

y^{^{''}} - 9 y^{^{'}} + 18 y = 0, y (0) = - 5, y^{^{'}} (0) = 2