Question

Evaluate ∫secx(5secx+8tanx) dx. Here C is the constant of integration.

Answer 1

We have the integral:

`\int secx(5secx+8tanx)dx`

We expand the expression:

`\int5sec^2x+8tanxsecxdx`

Using trigonometric identities we can rewrite:

`\int(5+8sinx)/(cos^2x)dx`

We expand again:

`\int(5)/(cos^2x)+(8sinx)/(cos^2x)dx`

We separate the integral as:

`\int(5)/(cos^2x)dx+\int(8sinx)/(cos^2x)dx`

By integration rules we have that:

`\int(5)/(cos^2x)dx=5tanx`

And

`8\int(sinx)/(cos^2x)dx=8secx`

All together:

`5tanx+8secx+C`

This would be the answer with the constant c.

`5tanx+8secx+C`