What is the integral of 2xsec^2(5x) ?

The integral of 2xsec^2(5x) is x^2sec^2(5x)-x^2tan^2(5x)+2/5 xtan(5x)+2/25 ln|cos(5x)|-x^2+C

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integral of 2xsec^2(5x)

\int 2 x sec^{2} (5 x) d x

x^{2} sec^{2} (5 x) - x^{2} tan^{2} (5 x) + \frac{2}{5} x tan (5 x) + \frac{2}{25} ln ∣ cos (5 x) ∣ - x^{2} + C

Solution steps

\int 2 x sec^{2} (5 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} (5 x) d x

Rewrite using trig identities

= 2 \cdot \int x (1 + tan^{2} (5 x)) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} sec^{2} (5 x) - \int 5 x^{2} sec^{2} (5 x) tan (5 x) d x

= 2 (\frac{1}{2} x^{2} sec^{2} (5 x) - \int 5 x^{2} sec^{2} (5 x) tan (5 x) d x)

\int 5 x^{2} sec^{2} (5 x) tan (5 x) d x = \frac{1}{50} (25 x^{2} tan^{2} (5 x) - 10 x tan (5 x) - 2 ln ∣ cos (5 x) ∣ + 25 x^{2})

= 2 (\frac{1}{2} x^{2} sec^{2} (5 x) - \frac{1}{50} (25 x^{2} tan^{2} (5 x) - 10 x tan (5 x) - 2 ln ∣ cos (5 x) ∣ + 25 x^{2}))

Simplify

2 (\frac{1}{2} x^{2} sec^{2} (5 x) - \frac{1}{50} (25 x^{2} tan^{2} (5 x) - 10 x tan (5 x) - 2 ln ∣ cos (5 x) ∣ + 25 x^{2})) : x^{2} sec^{2} (5 x) - x^{2} tan^{2} (5 x) + \frac{2}{5} x tan (5 x) + \frac{2}{25} ln ∣ cos (5 x) ∣ - x^{2}

= x^{2} sec^{2} (5 x) - x^{2} tan^{2} (5 x) + \frac{2}{5} x tan (5 x) + \frac{2}{25} ln ∣ cos (5 x) ∣ - x^{2}

Add a constant to the solution

= x^{2} sec^{2} (5 x) - x^{2} tan^{2} (5 x) + \frac{2}{5} x tan (5 x) + \frac{2}{25} ln ∣ cos (5 x) ∣ - x^{2} + C

\int x^{2} - 4 x + 5 d x

x \to 0 - lim (\frac{x}{ln ( x )})

\int \frac{1}{8 x + 5} d x

\frac{d}{d x} (ln (f (x)) \cdot f (x) (sin (x)))

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

What is the integral of 2xsec^2(5x) ?
The integral of 2xsec^2(5x) is x^2sec^2(5x)-x^2tan^2(5x)+2/5 xtan(5x)+2/25 ln|cos(5x)|-x^2+C