What is the integral of (2xsin(x))/(cos^3(x)) ?

The integral of (2xsin(x))/(cos^3(x)) is xtan^2(x)+x-tan(x)+C

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integral of (2xsin(x))/(cos^3(x))

\int \frac{2 x sin ( x )}{cos ^{3} ( x )} d x

x tan^{2} (x) + x - tan (x) + C

Solution steps

\int \frac{2 x sin ( x )}{cos ^{3} ( x )} d x

Take the constant out:

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

= 2 \cdot \int \frac{x sin ( x )}{cos ^{3} ( x )} d x

Rewrite using trig identities

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

Apply Integration By Parts

: \frac{1}{2} x tan^{2} (x) - \int \frac{1}{2} tan^{2} (x) d x

= 2 (\frac{1}{2} x tan^{2} (x) - \int \frac{1}{2} tan^{2} (x) d x)

\int \frac{1}{2} tan^{2} (x) d x = \frac{1}{2} (- x + tan (x))

= 2 (\frac{1}{2} x tan^{2} (x) - \frac{1}{2} (- x + tan (x)))

Simplify

2 (\frac{1}{2} x tan^{2} (x) - \frac{1}{2} (- x + tan (x))) : x tan^{2} (x) + x - tan (x)

= x tan^{2} (x) + x - tan (x)

Add a constant to the solution

= x tan^{2} (x) + x - tan (x) + C

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What is the integral of (2xsin(x))/(cos^3(x)) ?
The integral of (2xsin(x))/(cos^3(x)) is xtan^2(x)+x-tan(x)+C