What is the integral of (cos(sqrt(x))) ?

The integral of (cos(sqrt(x))) is 2(sqrt(x)sin(sqrt(x))+cos(sqrt(x)))+C

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integral of (cos(sqrt(x)))

\int (cos (x)) d x

2 (x sin (x) + cos (x)) + C

Solution steps

\int cos (x) d x

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 = x

= 2 (x sin (x) - (- cos (x)))

Simplify

= 2 (x sin (x) + cos (x))

Add a constant to the solution

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

x \to \frac{1}{4} lim (sin (\frac{π}{x}))

\frac{d}{d x} ((7 x + 1)^{2})

\int ln (x + (1 + x^{2})^{\frac{1}{2}}) d x

\int x y z^{2} d x

\int sec (x + 1) tan (x + 1) d x

What is the integral of (cos(sqrt(x))) ?
The integral of (cos(sqrt(x))) is 2(sqrt(x)sin(sqrt(x))+cos(sqrt(x)))+C