What is the integral of 1/(cos^2(5x+3)) ?

The integral of 1/(cos^2(5x+3)) is -3/5+1/5 tan(5x+3)+C

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integral of 1/(cos^2(5x+3))

\int \frac{1}{cos ^{2} ( 5 x + 3 )} d x

- \frac{3}{5} + \frac{1}{5} tan (5 x + 3) + C

Solution steps

\int \frac{1}{cos ^{2} ( 5 x + 3 )} d x

Use the basic trigonometric identity:

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

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

Rewrite using trig identities

= \int 1 + tan^{2} (5 x + 3) d x

Apply the Sum Rule:

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

= \int 1 d x + \int tan^{2} (5 x + 3) d x

\int 1 d x = x

\int tan^{2} (5 x + 3) d x = - x - \frac{3}{5} + \frac{1}{5} tan (5 x + 3)

= x - x - \frac{3}{5} + \frac{1}{5} tan (5 x + 3)

Simplify

= - \frac{3}{5} + \frac{1}{5} tan (5 x + 3)

Add a constant to the solution

= - \frac{3}{5} + \frac{1}{5} tan (5 x + 3) + C

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What is the integral of 1/(cos^2(5x+3)) ?
The integral of 1/(cos^2(5x+3)) is -3/5+1/5 tan(5x+3)+C