What is the integral of sec^2(5x+3) ?

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

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integral of sec^2(5x+3)

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

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

Solution steps

\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

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