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Integrate intcos^(3)xdx

Integrate $\int \cos ^{3} x d x$

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Power property of logarithms

Full solution

Q. Integrate

\int \cos ^{3} x d x

Identify Problem: Identify the integral that needs to be solved. $\newline$ We need to find the integral of $\cos^3(x)$ with respect to $x$ .
Rewrite Integral: Rewrite the integral in a more convenient form. $\newline$ We can use the trigonometric identity $\cos^2(x) = 1 - \sin^2(x)$ to rewrite $\cos^3(x)$ as $\cos(x) \cdot (1 - \sin^2(x))$ .
Set Up Expression: Set up the integral with the rewritten expression. $\newline$ The integral becomes $\int \cos(x) \cdot (1 - \sin^2(x)) \, dx$ .
Use Substitution: Use substitution to solve the integral. Let $u = \sin(x)$ , then $du = \cos(x) dx$ . The integral now becomes $\int(1 - u^2) du$ .
Integrate with u: Integrate with respect to $u$ . The integral of $1$ with respect to $u$ is $u$ , and the integral of $u^2$ with respect to $u$ is $(u^3)/3$ . So, $\int(1 - u^2) du = u - (u^3)/3 + C$ , where $C$ is the constant of integration.
Substitute Back: Substitute back in terms of $x$ . Since $u = \sin(x)$ , we substitute back to get the integral in terms of $x$ : $\sin(x) - (\sin^3(x))/3 + C$ .

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