What is the integral of 1/(cos^2(5x)) ?

The integral of 1/(cos^2(5x)) is 1/5 tan(5x)+C

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integral of 1/(cos^2(5x))

\int \frac{1}{cos ^{2} ( 5 x )} d x

\frac{1}{5} tan (5 x) + C

Solution steps

\int \frac{1}{cos ^{2} ( 5 x )} d x

Rewrite using trig identities

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

Apply u-substitution

: \int sec^{2} (u) \frac{1}{5} d u

= \int sec^{2} (u) \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int sec^{2} (u) d u

Use the common integral:

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

= \frac{1}{5} tan (u)

Substitute back

u = 5 x

= \frac{1}{5} tan (5 x)

Add a constant to the solution

= \frac{1}{5} tan (5 x) + C

x \to 1 lim (\frac{2 \cdot x + 3}{x + 4})

\int \frac{x + 1}{3 - 2 x - x ^{2}} d x

\frac{\partial}{\partial y} (e^{x y} tan (y))

\frac{d}{d x} (1 + \frac{5 e ^{- 2 x^{2}}}{2 - ∣ x ∣})

y = cos (sin (tan (7 x)))

What is the integral of 1/(cos^2(5x)) ?
The integral of 1/(cos^2(5x)) is 1/5 tan(5x)+C