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

The integral of (3x^2)/(sec(2x)) is 3(1/2 x^2sin(2x)+1/2 xcos(2x)-1/4 sin(2x))+C

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integral of (3x^2)/(sec(2x))

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

3 (\frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) - \frac{1}{4} sin (2 x)) + C

Solution steps

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

Take the constant out:

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

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

Use the basic trigonometric identity:

\frac{1}{sec ( x )} = cos (x)

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

Apply Integration By Parts

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

= 3 (\frac{1}{2} x^{2} sin (2 x) - \int x sin (2 x) d x)

\int x sin (2 x) d x = - \frac{1}{2} x cos (2 x) + \frac{1}{4} sin (2 x)

= 3 (\frac{1}{2} x^{2} sin (2 x) - (- \frac{1}{2} x cos (2 x) + \frac{1}{4} sin (2 x)))

Simplify

= 3 (\frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) - \frac{1}{4} sin (2 x))

Add a constant to the solution

= 3 (\frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) - \frac{1}{4} sin (2 x)) + C

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What is the integral of (3x^2)/(sec(2x)) ?
The integral of (3x^2)/(sec(2x)) is 3(1/2 x^2sin(2x)+1/2 xcos(2x)-1/4 sin(2x))+C