What is the integral of sec^2(7x+5) ?

The integral of sec^2(7x+5) is -5/7+1/7 tan(7x+5)+C

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integral of sec^2(7x+5)

\int sec^{2} (7 x + 5) d x

- \frac{5}{7} + \frac{1}{7} tan (7 x + 5) + C

Solution steps

\int sec^{2} (7 x + 5) d x

Rewrite using trig identities

= \int 1 + tan^{2} (7 x + 5) 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} (7 x + 5) d x

\int 1 d x = x

\int tan^{2} (7 x + 5) d x = - x - \frac{5}{7} + \frac{1}{7} tan (7 x + 5)

= x - x - \frac{5}{7} + \frac{1}{7} tan (7 x + 5)

Simplify

= - \frac{5}{7} + \frac{1}{7} tan (7 x + 5)

Add a constant to the solution

= - \frac{5}{7} + \frac{1}{7} tan (7 x + 5) + C

What is the integral of sec^2(7x+5) ?
The integral of sec^2(7x+5) is -5/7+1/7 tan(7x+5)+C