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

The integral of (sec(2a)x)^2 is (x^3sec^2(2a))/3+C

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integral of (sec(2a)x)^2

\int (sec (2 a) x)^{2} d x

\frac{x ^{3} sec ^{2} ( 2 a )}{3} + C

Solution steps

\int (sec (2 a) x)^{2} d x

Rewrite using trig identities

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

Take the constant out:

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

= \frac{1}{cos ^{2} ( 2 a )} \cdot \int x^{2} d x

Apply the Power Rule

: \frac{x ^{3}}{3}

= \frac{1}{cos ^{2} ( 2 a )} \cdot \frac{x ^{3}}{3}

Simplify

\frac{1}{cos ^{2} ( 2 a )} \cdot \frac{x ^{3}}{3} : \frac{x ^{3} sec ^{2} ( 2 a )}{3}

= \frac{x ^{3} sec ^{2} ( 2 a )}{3}

Add a constant to the solution

= \frac{x ^{3} sec ^{2} ( 2 a )}{3} + C

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What is the integral of (sec(2a)x)^2 ?
The integral of (sec(2a)x)^2 is (x^3sec^2(2a))/3+C