We have the integral:
[tex]\int secx(5secx+8tanx)dx[/tex]
We expand the expression:
[tex]\int5sec^2x+8tanxsecxdx[/tex]
Using trigonometric identities we can rewrite:
[tex]\int\frac{5+8sinx}{cos^2x}dx[/tex]
We expand again:
[tex]\int\frac{5}{cos^2x}+\frac{8sinx}{cos^2x}dx[/tex]
We separate the integral as:
[tex]\int\frac{5}{cos^2x}dx+\int\frac{8sinx}{cos^2x}dx[/tex]
By integration rules we have that:
[tex]\int\frac{5}{cos^2x}dx=5tanx[/tex]
And
[tex]8\int\frac{sinx}{cos^2x}dx=8secx[/tex]
All together:
[tex]5tanx+8secx+C[/tex]
This would be the answer with the constant c.
[tex]5tanx+8secx+C[/tex]
