What is the integral of sec(5x+4)tan(5x+4) ?

The integral of sec(5x+4)tan(5x+4) is 1/5 sin(5x+4)tan(5x+4)+1/5 cos(5x+4)+C

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integral of sec(5x+4)tan(5x+4)

\int sec (5 x + 4) tan (5 x + 4) d x

\frac{1}{5} sin (5 x + 4) tan (5 x + 4) + \frac{1}{5} cos (5 x + 4) + C

Solution steps

\int sec (5 x + 4) tan (5 x + 4) d x

Rewrite using trig identities

= \int \frac{sin ( 5 x + 4 )}{cos ^{2} ( 5 x + 4 )} d x

Apply Integration By Parts

: sin (5 x + 4) (- \frac{4}{5} + \frac{1}{5} tan (5 x + 4)) - \int - 4 cos (5 x + 4) + sin (4 + 5 x) d x

= sin (5 x + 4) (- \frac{4}{5} + \frac{1}{5} tan (5 x + 4)) - \int - 4 cos (5 x + 4) + sin (4 + 5 x) d x

\int - 4 cos (5 x + 4) + sin (4 + 5 x) d x = - \frac{4}{5} sin (5 x + 4) - \frac{1}{5} cos (4 + 5 x)

= sin (5 x + 4) (- \frac{4}{5} + \frac{1}{5} tan (5 x + 4)) - (- \frac{4}{5} sin (5 x + 4) - \frac{1}{5} cos (4 + 5 x))

Simplify

sin (5 x + 4) (- \frac{4}{5} + \frac{1}{5} tan (5 x + 4)) - (- \frac{4}{5} sin (5 x + 4) - \frac{1}{5} cos (4 + 5 x)) : \frac{1}{5} sin (5 x + 4) tan (5 x + 4) + \frac{1}{5} cos (5 x + 4)

= \frac{1}{5} sin (5 x + 4) tan (5 x + 4) + \frac{1}{5} cos (5 x + 4)

Add a constant to the solution

= \frac{1}{5} sin (5 x + 4) tan (5 x + 4) + \frac{1}{5} cos (5 x + 4) + C

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What is the integral of sec(5x+4)tan(5x+4) ?
The integral of sec(5x+4)tan(5x+4) is 1/5 sin(5x+4)tan(5x+4)+1/5 cos(5x+4)+C