### Home > APCALC > Chapter 7 > Lesson 7.4.4 > Problem7-221

7-221.

Multiple Choice: $\int \operatorname { cos } \sqrt { x } d x =$

 $\sin\sqrt { x }+C$ $2x \sin(x) - 2\sin(x) + C$ $2\sqrt { x }\sin \sqrt { x }+ 2 \cos\sqrt { x }+C$ $2x \cos(x) - 2\sin(x) + C$ $2\sqrt{x}\cos\sqrt{x} + 2 \cos\sqrt{x} + C$

First use $u$-substitution:

Then use integration by parts:

$U=\sqrt{x}$

$\frac{dU}{dx}=\frac{1}{2\sqrt{x}}$

$2\sqrt{x}dU=dx=2UdU$

$\int 2U\text{cos}(U)dU$

Let $f(x) = 2U$ and $dg = \cos(U)dU$.
Then $df =$ __________ and $g\left(x\right) =$ ____________.

$\text{Evaluate: }f(x)g(x)-\int g(x)df$