What is the integral of ye^xcos(yz) ?

The integral of ye^xcos(yz) is e^xsin(yz)+C

integral of ye^xcos(yz)

\int y e^{x} cos (yz) d z

e^{x} sin (yz) + C

Solution steps

\int y e^{x} cos (yz) d z

Take the constant out:

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

= y e^{x} \cdot \int cos (yz) d z

Apply u-substitution

= y e^{x} \cdot \int \frac{cos ( u )}{y} d u

Take the constant out:

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

= y e^{x} \frac{1}{y} \cdot \int cos (u) d u

Use the common integral:

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

= y e^{x} \frac{1}{y} sin (u)

Substitute back

u = yz

= y e^{x} \frac{1}{y} sin (yz)

Simplify

y e^{x} \frac{1}{y} sin (yz) : e^{x} sin (yz)

= e^{x} sin (yz)

Add a constant to the solution

= e^{x} sin (yz) + C

\int a + b x^{2} d x

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

\int \frac{e ^{2 x}}{e ^{2 x} - 8} d x

\int \frac{2 x ^{2} - 4 x - 3}{x ^{2}} d x

in t e g r a l tan^{2} (x) sec^{2} (x)

What is the integral of ye^xcos(yz) ?
The integral of ye^xcos(yz) is e^xsin(yz)+C