int(cos sqrtx)/(sqrtx)dx=

Substitution step: Let u = sqrt(x), which implies that x = u^2 and dx = 2u du. This substitution will simplify the integral.Rewrite in terms of u: Rewrite the integral in terms of u. The integral becomes int(cos(u))/(u) * 2u du, where we have replaced sqrt(x) with u and dx with 2u du.Simplify the integral: Simplify the integral by canceling the u in the denominator with one of the u's in the numerator. The integral now is 2 * int(cos(u)) du.Integrate cos(u): Integrate 2 * int(cos(u)) du with respect to u. The integral of cos(u) with respect to u is sin(u).Multiply by 2: Multiply the result of the integration by 2. The result is 2sin(u) + C, where C is the constant of integration.Substitute back x: Substitute back the original variable x into the result. Since u = sqrt(x), the result is 2sin(sqrt(x)) + C.

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Find indefinite integrals using the substitution and by parts

Full solution

\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x

Substitution step: Let $u = \sqrt{x}$ , which implies that $x = u^2$ and $dx = 2u \, du$ . This substitution will simplify the integral.
Rewrite in terms of $u$ : Rewrite the integral in terms of $u$ . The integral becomes $\int\frac{\cos(u)}{u} \cdot 2u \, du$ , where we have replaced $\sqrt{x}$ with $u$ and $dx$ with $2u \, du$ .
Simplify the integral: Simplify the integral by canceling the $u$ in the denominator with one of the $u$ 's in the numerator. The integral now is $2 \times \int(\cos(u)) \, du$ .
Integrate $\cos(u)$ : Integrate $2 \times \int(\cos(u)) \, du$ with respect to $u$ . The integral of $\cos(u)$ with respect to $u$ is $\sin(u)$ .
Multiply by $2$ : Multiply the result of the integration by $2$ . The result is $2\sin(u) + C$ , where $C$ is the constant of integration.
Substitute back $x$ : Substitute back the original variable $x$ into the result. Since $u = \sqrt{x}$ , the result is $2\sin(\sqrt{x}) + C$ .