How do you use Integration by Substitution to find $\int \frac{cos (\sqrt{x})}{\sqrt{x}} d x$ $intcos(sqrt(x))/sqrt(x)dx$ ?

Question

2821 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2014-09-26T09:13:10

Begin by making the $u$ $u$ -substitution.

Let $u = \sqrt{x}$ $u=sqrt(x)$

$u = x^{\frac{1}{2}}$ $u=x^(1/2)$

$d u = \frac{1}{2} \cdot x^{\frac{- 1}{2}} d x$ $du= 1/2*x^((-1)/2) dx$

$d u = \frac{1}{2 \cdot x^{\frac{1}{2}}} d x$ $du= 1/(2*x^((1)/2)) dx$

$2 d u = \frac{d x}{x^{\frac{1}{2}}}$ $2du= (dx)/(x^((1)/2))$

$2 d u = \frac{d x}{\sqrt{x}} \Rightarrow$ $2du= (dx)/sqrt(x) =>$ Notice that is can be substituted in the integral

$\int \frac{cos \sqrt{x}}{\sqrt{x}} d x = \int cos (u) \cdot 2 d u = 2 \int cos (u) d u$ $int(cos sqrt(x))/(sqrt(x))dx=int cos(u)*2du=2intcos(u)du$

$= 2 [sin (u)] + C$ $=2[sin(u)]+C$

Rewrite in terms of $x$ $x$

$= 2 [sin (\sqrt{x})] + C$ $=2[sin(sqrt(x))]+C$

How do you use Integration by Substitution to find ∫cos(√x)√xdxintcos(sqrt(x))/sqrt(x)dx?