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

The integral of x/(cos^2(3x)) is 1/2 x^2sec^2(3x)-1/18 (9x^2tan^2(3x)-6xtan(3x)-2ln|cos(3x)|+9x^2)+C

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

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

\frac{1}{2} x^{2} sec^{2} (3 x) - \frac{1}{18} (9 x^{2} tan^{2} (3 x) - 6 x tan (3 x) - 2 ln ∣ cos (3 x) ∣ + 9 x^{2}) + C

Solution steps

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

Use the basic trigonometric identity:

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

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

Rewrite using trig identities

= \int x (1 + tan^{2} (3 x)) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} sec^{2} (3 x) - \int 3 x^{2} sec^{2} (3 x) tan (3 x) d x

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

\int 3 x^{2} sec^{2} (3 x) tan (3 x) d x = \frac{1}{18} (9 x^{2} tan^{2} (3 x) - 6 x tan (3 x) - 2 ln ∣ cos (3 x) ∣ + 9 x^{2})

= \frac{1}{2} x^{2} sec^{2} (3 x) - \frac{1}{18} (9 x^{2} tan^{2} (3 x) - 6 x tan (3 x) - 2 ln ∣ cos (3 x) ∣ + 9 x^{2})

Add a constant to the solution

= \frac{1}{2} x^{2} sec^{2} (3 x) - \frac{1}{18} (9 x^{2} tan^{2} (3 x) - 6 x tan (3 x) - 2 ln ∣ cos (3 x) ∣ + 9 x^{2}) + C

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

\int (3 x^{2} + 5) x^{3} + 5 x d x

\int \frac{3 - x ^{2}}{x} d x

\int (x)^{3} d x

\int \frac{1}{4 - t ^{2}} d t

What is the integral of x/(cos^2(3x)) ?
The integral of x/(cos^2(3x)) is 1/2 x^2sec^2(3x)-1/18 (9x^2tan^2(3x)-6xtan(3x)-2ln|cos(3x)|+9x^2)+C