What is the integral of ae^{3x} ?

The integral of ae^{3x} is a 1/3 e^{3x}+C

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integral of ae^{3x}

\int a e^{3 x} d x

a \frac{1}{3} e^{3 x} + C

Solution steps

\int a e^{3 x} d x

Take the constant out:

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

= a \cdot \int e^{3 x} d x

Apply u-substitution

: \int e^{u} \frac{1}{3} d u

= a \cdot \int e^{u} \frac{1}{3} d u

Take the constant out:

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

= a \frac{1}{3} \cdot \int e^{u} d u

Use the common integral:

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

= a \frac{1}{3} e^{u}

Substitute back

u = 3 x

= a \frac{1}{3} e^{3 x}

Add a constant to the solution

= a \frac{1}{3} e^{3 x} + C

x \to - 2 lim (x^{2} + 2 x + 4)

f (x) = x^{6} - 4 x^{4} + x + 1

\frac{d y}{d x} = \frac{1}{x} (y^{2} + 0.6 y)

\int_{- 1}^{3} - x^{2} + 2 x + 3 d x

f^{^{'}} (x) = x (3 + 5 x), f (1) = 7