What is the integral of sqrt(e^t) ?

The integral of sqrt(e^t) is 2e^{t/2}+C

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integral of sqrt(e^t)

\int e^{t} d t

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

Solution steps

\int e^{t} d t

Apply radical rule:

assuming

a \geq 0

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

Apply u-substitution

: \int e^{u} \cdot 2 d u

= \int e^{u} \cdot 2 d u

Take the constant out:

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

= 2 \cdot \int e^{u} d u

Use the common integral:

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

= 2 e^{u}

Substitute back

u = \frac{t}{2}

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

Add a constant to the solution

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

y^{^{''}} + 2 y^{^{'}} + y = 0, y (0) = 0, y^{^{'}} (0) = 1

x y^{^{'}} + y = x e^{x}

\frac{d}{d x} (c_{1} e^{- x} + c_{2} x e^{- x})

\int cot (d) x d x

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