Question

integrate cos root x / root x

Answer 1

2 sin(√x) + C

Answer 2

To integrate the given function, we use the method of substitution.

Let the given integral be $I = \int \frac{\cos(\sqrt{x})}{\sqrt{x}} dx$.

Let's make a substitution: Let $u = \sqrt{x}$.

Now, we need to find $du$ in terms of $dx$. Differentiate $u$ with respect to $x$: $\frac{du}{dx} = \frac{d}{dx}(x^{1/2})$ $\frac{du}{dx} = \frac{1}{2}x^{(1/2 - 1)}$ $\frac{du}{dx} = \frac{1}{2}x^{-1/2}$ $\frac{du}{dx} = \frac{1}{2\sqrt{x}}$

Now, rearrange to solve for $dx/\sqrt{x}$: $2 du = \frac{1}{\sqrt{x}} dx$

Substitute $u = \sqrt{x}$ and $2 du = \frac{1}{\sqrt{x}} dx$ into the integral: $I = \int \cos(u) \cdot (2 du)$ $I = 2 \int \cos(u) du$

Now, integrate with respect to $u$: The integral of $\cos(u)$ is $\sin(u)$. $I = 2 \sin(u) + C$

Finally, substitute back $u = \sqrt{x}$: $I = 2 \sin(\sqrt{x}) + C$

Where $C$ is the constant of integration.

The final answer is $\boxed{2 \sin(\sqrt{x}) + C}$.

Explanation of the solution: The integral $\int \frac{\cos(\sqrt{x})}{\sqrt{x}} dx$ is solved using substitution. Let $u = \sqrt{x}$. Differentiating gives $du = \frac{1}{2\sqrt{x}} dx$, which means $2 du = \frac{1}{\sqrt{x}} dx$. Substituting these into the integral yields $2 \int \cos(u) du$. Integrating $\cos(u)$ gives $\sin(u)$, so the result is $2 \sin(u) + C$. Substituting back $u = \sqrt{x}$, the final answer is $2 \sin(\sqrt{x}) + C$.