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

The integral of xtan^2(3x^2) is 1/2 (-x^2+1/3 tan(3x^2))+C

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integral of xtan^2(3x^2)

\int x tan^{2} (3 x^{2}) d x

\frac{1}{2} (- x^{2} + \frac{1}{3} tan (3 x^{2})) + C

Solution steps

\int x tan^{2} (3 x^{2}) d x

Apply u-substitution

: \int \frac{tan ^{2} ( 3 u )}{2} d u

= \int \frac{tan ^{2} ( 3 u )}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int tan^{2} (3 u) d u

Rewrite using trig identities

= \frac{1}{2} \cdot \int - 1 + sec^{2} (3 u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (- \int 1 d u + \int sec^{2} (3 u) d u)

\int 1 d u = u

\int sec^{2} (3 u) d u = \frac{1}{3} tan (3 u)

= \frac{1}{2} (- u + \frac{1}{3} tan (3 u))

Substitute back

u = x^{2}

= \frac{1}{2} (- x^{2} + \frac{1}{3} tan (3 x^{2}))

Add a constant to the solution

= \frac{1}{2} (- x^{2} + \frac{1}{3} tan (3 x^{2})) + C

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What is the integral of xtan^2(3x^2) ?
The integral of xtan^2(3x^2) is 1/2 (-x^2+1/3 tan(3x^2))+C