Finding Indefinite Integrals ...

Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ cos³ 𝓍/2 d𝓍

Accepted Answer

Welcome back everyone. Compute the indefinite integral of sine cubed of x divided by 3 d x. For this problem we can begin by decomposing sine cubed. We can essentially rewrite sine cubed. Of x divided by 3 s sin squared of x divided by 3 multiplied by sin of x divided by 3. From here we are going to use the Pythagorean identity. Specifically, if we consider sine squared, we can rewrite sin squared off x divided by 3s 1 minus cosine squared of x divided by 3. So now the integrant is going to be sin cubed of x divided by 3, which is 1 minus cosine squared of x divided by 3 multiplied by sin of x divided by 3, and from here we can use substitution. Let's suppose that u is equal to cosine of x divided by 3. Now the next step is to find the derivative of u with respect to x. So that's the derivative of cosine of x divided by 3, which is negative sin of x divided by 3 multiplied by the derivative of x divided by 3, and that's 1/3. So now what we can do is simply rearrange this expression, right? And we can show that. Sine of x divided by 3 multiplied by d x. is equal to duu divided by 1/3, which is -3DU. Which allows us to use u substitution and now the integral becomes integral of sin squared of x divided by 3 multiplied by sin of x divided by 3 d x which is now equal to. Integral of. 1 minus cosine squared of x divided by 3 multiplied by a sin of x divided by 3 d x and now using the variables we get integral of 1 minus u squared multiplied by -3 duu we can factor out -3 and we are ready to integrate. So we are going to get -3 in parentheses. U minus u cubed divided by 3 using the power rule plus a constant of integration c. Now let's go ahead and distribute. That's -3 U. Plus you cubed. Plus C. And finally, we're going to replace you with cosines, so we're going to get -3. Cosine of x divided by 3 plus. Cosine cubed of x divided by 3 plus a constant of integration c and that's our final answer for this problem. Thank you for watching.

Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation. ∫ cos³ 𝓍/2 d𝓍

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Finding Indefinite Integrals
Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ cos³ 𝓍/2 d𝓍