What is the integral of 6^tsin(6^t) ?

The integral of 6^tsin(6^t) is -1/(ln(6))cos(6^t)+C

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integral of 6^tsin(6^t)

\int 6^{t} sin (6^{t}) d t

- \frac{1}{ln ( 6 )} cos (6^{t}) + C

Solution steps

\int 6^{t} sin (6^{t}) d t

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = 6^{t}

= \frac{1}{ln ( 6 )} (- cos (6^{t}))

Simplify

= - \frac{1}{ln ( 6 )} cos (6^{t})

Add a constant to the solution

= - \frac{1}{ln ( 6 )} cos (6^{t}) + C

What is the integral of 6^tsin(6^t) ?
The integral of 6^tsin(6^t) is -1/(ln(6))cos(6^t)+C