What is the integral of 3e^{3x}sin(e^{3x}) ?

The integral of 3e^{3x}sin(e^{3x}) is -cos(e^{3x})+C

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integral of 3e^{3x}sin(e^{3x})

\int 3 e^{3 x} sin (e^{3 x}) d x

- cos (e^{3 x}) + C

Solution steps

\int 3 e^{3 x} sin (e^{3 x}) d x

Take the constant out:

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

= 3 \cdot \int e^{3 x} sin (e^{3 x}) d x

Apply u-substitution

: \int \frac{sin ( u )}{3} d u

= 3 \cdot \int \frac{sin ( u )}{3} d u

Take the constant out:

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

= 3 \cdot \frac{1}{3} \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= 3 \cdot \frac{1}{3} (- cos (u))

Substitute back

u = e^{3 x}

= 3 \cdot \frac{1}{3} (- cos (e^{3 x}))

Simplify

3 \cdot \frac{1}{3} (- cos (e^{3 x})) : - cos (e^{3 x})

= - cos (e^{3 x})

Add a constant to the solution

= - cos (e^{3 x}) + C

f (x) = x x

(4 x^{2} + 2 x y) \frac{d y}{d x} + (12 x y + 2 y^{2}) = 0

\int_{\frac{1}{2}}^{1} sin (2 π x) d x

y = - x^{2} + 4, y = 2 x + 1

500000 + 5000 t^{2}

What is the integral of 3e^{3x}sin(e^{3x}) ?
The integral of 3e^{3x}sin(e^{3x}) is -cos(e^{3x})+C