What is the integral of x((sin(x))/(cos(x)))^2 ?

The integral of x((sin(x))/(cos(x)))^2 is -x^2+xtan(x)+(x^2)/2+ln|cos(x)|+C

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integral of x((sin(x))/(cos(x)))^2

\int x (\frac{sin ( x )}{cos ( x )})^{2} d x

- x^{2} + x tan (x) + \frac{x ^{2}}{2} + ln ∣ cos (x) ∣ + C

Solution steps

\int x (\frac{sin ( x )}{cos ( x )})^{2} d x

Simplify

x (\frac{sin ( x )}{cos ( x )})^{2} : x tan^{2} (x)

= \int x tan^{2} (x) d x

Apply Integration By Parts

= x (- x + tan (x)) - \int - x + tan (x) d x

\int - x + tan (x) d x = - \frac{x ^{2}}{2} - ln ∣ cos (x) ∣

= x (- x + tan (x)) - (- \frac{x ^{2}}{2} - ln ∣ cos (x) ∣)

Simplify

x (- x + tan (x)) - (- \frac{x ^{2}}{2} - ln ∣ cos (x) ∣) : - x^{2} + x tan (x) + \frac{x ^{2}}{2} + ln ∣ cos (x) ∣

= - x^{2} + x tan (x) + \frac{x ^{2}}{2} + ln ∣ cos (x) ∣

Add a constant to the solution

= - x^{2} + x tan (x) + \frac{x ^{2}}{2} + ln ∣ cos (x) ∣ + C

What is the integral of x((sin(x))/(cos(x)))^2 ?
The integral of x((sin(x))/(cos(x)))^2 is -x^2+xtan(x)+(x^2)/2+ln|cos(x)|+C