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

The integral of sec(2x+5)tan(2x+5) is 1/2 cos(5+2x)+1/2 sin(5+2x)tan(5+2x)+C

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integral of sec(2x+5)tan(2x+5)

\int sec (2 x + 5) tan (2 x + 5) d x

\frac{1}{2} cos (5 + 2 x) + \frac{1}{2} sin (5 + 2 x) tan (5 + 2 x) + C

Solution steps

\int sec (2 x + 5) tan (2 x + 5) d x

Rewrite using trig identities

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

Apply Integration By Parts

: - \int sin (5 + 2 x) d x + \frac{1}{2} sin (5 + 2 x) tan (5 + 2 x)

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

\int sin (5 + 2 x) d x = - \frac{1}{2} cos (5 + 2 x)

= - (- \frac{1}{2} cos (5 + 2 x)) + \frac{1}{2} sin (5 + 2 x) tan (5 + 2 x)

Simplify

= \frac{1}{2} cos (5 + 2 x) + \frac{1}{2} sin (5 + 2 x) tan (5 + 2 x)

Add a constant to the solution

= \frac{1}{2} cos (5 + 2 x) + \frac{1}{2} sin (5 + 2 x) tan (5 + 2 x) + C

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What is the integral of sec(2x+5)tan(2x+5) ?
The integral of sec(2x+5)tan(2x+5) is 1/2 cos(5+2x)+1/2 sin(5+2x)tan(5+2x)+C