What is the integral of (sin(x)e^{i*t}) ?

The integral of (sin(x)e^{i*t}) is sin(x)sin(t)-isin(x)cos(t)+C

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integral of (sin(x)e^{i*t})

\int (sin (x) e^{i \cdot t}) d t

sin (x) sin (t) - i sin (x) cos (t) + C

Solution steps

\int sin (x) e^{i t} d t

Take the constant out:

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

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

Apply u-substitution

: \int - i e^{u} d u

= sin (x) \cdot \int - i e^{u} d u

Take the constant out:

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

= sin (x) (- i \cdot \int e^{u} d u)

Use the common integral:

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

= sin (x) (- i e^{u})

Substitute back

u = i t

= sin (x) (- i e^{i t})

Simplify

sin (x) (- i e^{i t}) : sin (x) sin (t) - i sin (x) cos (t)

= sin (x) sin (t) - i sin (x) cos (t)

Add a constant to the solution

= sin (x) sin (t) - i sin (x) cos (t) + C

\frac{d}{d x} (ln (x) ln (x) ln (x))

x^{\frac{1}{7}}

\frac{d}{d y} ((x + y) e^{y})

\frac{d}{d x} ((4 + 8 x) cos (1 - x))

\frac{d}{d x} ((x + 1) (x^{3} + 3))

What is the integral of (sin(x)e^{i*t}) ?
The integral of (sin(x)e^{i*t}) is sin(x)sin(t)-isin(x)cos(t)+C