What is the integral of 1/(sec(sqrt(y))) ?

The integral of 1/(sec(sqrt(y))) is 2(sqrt(y)sin(sqrt(y))+cos(sqrt(y)))+C

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integral of 1/(sec(sqrt(y)))

\int \frac{1}{sec ( y )} d y

2 (y sin (y) + cos (y)) + C

Solution steps

\int \frac{1}{sec ( y )} d y

Use the basic trigonometric identity:

\frac{1}{sec ( x )} = cos (x)

= \int cos (y) d y

Apply u-substitution

: \int cos (u) \cdot 2 u d u

= \int cos (u) \cdot 2 u d u

Take the constant out:

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

= 2 \cdot \int cos (u) u d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= 2 (u sin (u) - \int sin (u) d u)

\int sin (u) d u = - cos (u)

= 2 (u sin (u) - (- cos (u)))

Substitute back

u = y

= 2 (y sin (y) - (- cos (y)))

Simplify

= 2 (y sin (y) + cos (y))

Add a constant to the solution

= 2 (y sin (y) + cos (y)) + C

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\int \frac{1}{8 - 3 x} d x

What is the integral of 1/(sec(sqrt(y))) ?
The integral of 1/(sec(sqrt(y))) is 2(sqrt(y)sin(sqrt(y))+cos(sqrt(y)))+C