What is the integral of the function f(x)=sin 2x? (-1//2)cos x+C (1//2)sin x+C (-1//2)cos 2x+C (1//2)sin 2x+C

Identify Integral: Identify the integral that needs to be solved. We need to find the integral of f(x) = sin(2x).Substitution Method: Use the substitution method for integration. Let u = 2x, which implies that du/dx = 2 or du = 2dx. Therefore, dx = du/2.Rewrite in terms of u: Rewrite the integral in terms of u. The integral of sin(2x)dx becomes (1/2)∫sin(u)du after substituting dx with du/2.Integrate sin(u): Integrate sin(u) with respect to u. The integral of sin(u) with respect to u is -cos(u) + C.Substitute back x: Substitute back the original variable x into the integral. Since u = 2x, the integral becomes (-1/2)cos(2x) + C.Write Final Answer: Write the final answer. The integral of the function f(x) = sin(2x) is (-1/2)cos(2x) + C.

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What is the integral of the function f(x)=sin 2x?

(-1//2)cos x+C

(1//2)sin x+C

(-1//2)cos 2x+C

(1//2)sin 2x+C

What is the integral of the function $f(x) = \sin(2x)$ ? $\newline$ $\frac{-1}{2}\cos(x) + C$ $\newline$ $\frac{1}{2}\sin(x) + C$ $\newline$ $\frac{-1}{2}\cos(2x) + C$ $\newline$ $\frac{1}{2}\sin(2x) + C$

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Calculus

Evaluate definite integrals using the chain rule

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Q. What is the integral of the function

f(x) = \sin(2x)

\newline

\frac{-1}{2}\cos(x) + C

\newline

\frac{1}{2}\sin(x) + C

\newline

\frac{-1}{2}\cos(2x) + C

\newline

\frac{1}{2}\sin(2x) + C

Identify Integral: Identify the integral that needs to be solved. $\newline$ We need to find the integral of $f(x) = \sin(2x)$ .
Substitution Method: Use the substitution method for integration. $\newline$ Let $u = 2x$ , which implies that $\frac{du}{dx} = 2$ or $du = 2dx$ . Therefore, $dx = \frac{du}{2}$ .
Rewrite in terms of $u$ : Rewrite the integral in terms of $u$ . The integral of $\sin(2x)\,dx$ becomes $(1/2)\int \sin(u)\,du$ after substituting $dx$ with $du/2$ .
Integrate $\sin(u)$ : Integrate $\sin(u)$ with respect to $u$ . The integral of $\sin(u)$ with respect to $u$ is $-\cos(u) + C$ .
Substitute back $x$ : Substitute back the original variable $x$ into the integral. $\newline$ Since $u = 2x$ , the integral becomes $(-\frac{1}{2})\cos(2x) + C$ .
Write Final Answer: Write the final answer. $\newline$ The integral of the function $f(x) = \sin(2x)$ is $(-\frac{1}{2})\cos(2x) + C$ .