What is the integral of 6e^{6x}sin(e^{6x}) ?

The integral of 6e^{6x}sin(e^{6x}) is -cos(e^{6x})+C

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integral of 6e^{6x}sin(e^{6x})

\int 6 e^{6 x} sin (e^{6 x}) d x

- cos (e^{6 x}) + C

Solution steps

\int 6 e^{6 x} sin (e^{6 x}) d x

Take the constant out:

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

= 6 \cdot \int e^{6 x} sin (e^{6 x}) d x

Apply u-substitution

: \int \frac{sin ( u )}{6} d u

= 6 \cdot \int \frac{sin ( u )}{6} d u

Take the constant out:

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

= 6 \cdot \frac{1}{6} \cdot \int sin (u) d u

Use the common integral:

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

= 6 \cdot \frac{1}{6} (- cos (u))

Substitute back

u = e^{6 x}

= 6 \cdot \frac{1}{6} (- cos (e^{6 x}))

Simplify

6 \cdot \frac{1}{6} (- cos (e^{6 x})) : - cos (e^{6 x})

= - cos (e^{6 x})

Add a constant to the solution

= - cos (e^{6 x}) + C

\int \frac{1}{4 y ^{2}} d y

\frac{x + 1}{x + 3}

y = (2 x^{2} + 3) (4 x - 3)

\frac{x}{6 + x}

9 x - 2 x^{2}

What is the integral of 6e^{6x}sin(e^{6x}) ?
The integral of 6e^{6x}sin(e^{6x}) is -cos(e^{6x})+C