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

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

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

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

5 sin (4 + sec (x)) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int sec (x) tan (x) cos (4 + sec (x)) d x

Apply u-substitution

= 5 \cdot \int cos (4 + u) d u

Apply u-substitution

= 5 \cdot \int cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= 5 sin (v)

Substitute back

= 5 sin (4 + sec (x))

Add a constant to the solution

= 5 sin (4 + sec (x)) + C

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