What is the integral of x^3tan(xy) ?

The integral of x^3tan(xy) is -x^2ln|cos(xy)|+C

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integral of x^3tan(xy)

\int x^{3} tan (x y) d y

- x^{2} ln ∣ cos (x y) ∣ + C

Solution steps

\int x^{3} tan (x y) d y

Take the constant out:

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

= x^{3} \cdot \int tan (x y) d y

Rewrite using trig identities

= x^{3} \cdot \int \frac{sin ( x y )}{cos ( x y )} d y

Apply u-substitution

= x^{3} \cdot \int - \frac{1}{xu} d u

Take the constant out:

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

= x^{3} (- \frac{1}{x} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= x^{3} (- \frac{1}{x} ln ∣ u ∣)

Substitute back

u = cos (x y)

= x^{3} (- \frac{1}{x} ln ∣ cos (x y) ∣)

Simplify

x^{3} (- \frac{1}{x} ln ∣ cos (x y) ∣) : - x^{2} ln ∣ cos (x y) ∣

= - x^{2} ln ∣ cos (x y) ∣

Add a constant to the solution

= - x^{2} ln ∣ cos (x y) ∣ + C

What is the integral of x^3tan(xy) ?
The integral of x^3tan(xy) is -x^2ln|cos(xy)|+C