What is the integral of 2^xsin(2^x) ?

The integral of 2^xsin(2^x) is -1/(ln(2))cos(2^x)+C

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integral of 2^xsin(2^x)

\int 2^{x} sin (2^{x}) d x

- \frac{1}{ln ( 2 )} cos (2^{x}) + C

Solution steps

\int 2^{x} sin (2^{x}) d x

Apply u-substitution

: \int \frac{sin ( u )}{ln ( 2 )} d u

= \int \frac{sin ( u )}{ln ( 2 )} d u

Take the constant out:

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

= \frac{1}{ln ( 2 )} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{ln ( 2 )} (- cos (u))

Substitute back

u = 2^{x}

= \frac{1}{ln ( 2 )} (- cos (2^{x}))

Simplify

= - \frac{1}{ln ( 2 )} cos (2^{x})

Add a constant to the solution

= - \frac{1}{ln ( 2 )} cos (2^{x}) + C

\int \frac{1}{- x ^{2} - 2 x + 3} d x

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

\int (x - x^{2}) d x

What is the integral of 2^xsin(2^x) ?
The integral of 2^xsin(2^x) is -1/(ln(2))cos(2^x)+C