What is the integral of (cos(y))/(x+3) ?

The integral of (cos(y))/(x+3) is cos(y)ln|3+x|+C

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integral of (cos(y))/(x+3)

\int \frac{cos ( y )}{x + 3} d x

cos (y) ln ∣ 3 + x ∣ + C

Solution steps

\int \frac{cos ( y )}{x + 3} d x

Take the constant out:

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

= cos (y) \cdot \int \frac{1}{3 + x} d x

Apply u-substitution

: \int \frac{1}{u} d u

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= cos (y) ln ∣ u ∣

Substitute back

u = 3 + x

= cos (y) ln ∣ 3 + x ∣

Add a constant to the solution

= cos (y) ln ∣ 3 + x ∣ + C

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What is the integral of (cos(y))/(x+3) ?
The integral of (cos(y))/(x+3) is cos(y)ln|3+x|+C