What is the integral of xcos(xz) ?

The integral of xcos(xz) is sin(xz)+C

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integral of xcos(xz)

\int x cos (x z) d z

sin (x z) + C

Solution steps

\int x cos (x z) d z

Take the constant out:

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

= x \cdot \int cos (x z) d z

Apply u-substitution

: \int \frac{cos ( u )}{x} d u

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

Take the constant out:

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

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

Use the common integral:

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

= x \frac{1}{x} sin (u)

Substitute back

u = x z

= x \frac{1}{x} sin (x z)

Simplify

x \frac{1}{x} sin (x z) : sin (x z)

= sin (x z)

Add a constant to the solution

= sin (x z) + C

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