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

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

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

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

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

Solution steps

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

Rewrite using trig identities

= 2 \cdot \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

= 2 (\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})

= 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}))

Simplify

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})) : x^{2} sec^{2} (3 x) - x^{2} tan^{2} (3 x) + \frac{2}{3} x tan (3 x) + \frac{2}{9} ln ∣ cos (3 x) ∣ - x^{2}

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

Add a constant to the solution

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

x \to 0 lim (e^{2 π})

\int e^{x} x d x

\frac{1}{( s - 3 ) ^{2} + 3}

\int \frac{35 x ^{4} - 3}{7 x ^{5} - 3 x} d x

(- 7, - 3), (- 5, - 8)

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