Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following...

Question

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.

∫ 𝓍 cos²𝓍² d𝓍

Accepted Answer

Welcome back, everyone. Determine the integral of x sine squared of X 2 DX. For this problem, let's begin with U substitution and let's suppose that u equals X2. Then the derivative DU divided by DX is the derivative of X2, and this is equal to 2 X. Let's notice that our integrant contains XDX. So we can rearrange this expression and say that X multiplied by DX is DU divided by 2, or simply 1/2 DU. So our integral can be rewritten as sine squared of you. Multiplied by 1/2 the U or simply d u and then we can factor out the constant which is 1/2. So now we have a new integral, one half integral of sine squared. UDU, and what we can do is simply recall the half angle formula. Let's recall that sine squared of U can be rewritten as a 1 minus cosine. Of 2 you all divided by 2. Meaning for our integral we can write one half. Integral of 1 minus cosine of 2 U. Divided by 2 D. Now, let's go ahead and split our integral into two parts. We have 1/2 in C, so our first integral is 1/2 the U. And then we're subtracting integral of cosine of 2 you divided by 2. The EU Now let's notice that each integral contains a constant, and that's 1/2, so we can factor it out, and 1/2 multiplied by 1/2 is 1/4, so we have a new constant which is 1/4. In Cs, our first integral becomes integral of the U. And then we're subtracting integral of cosine. Of to you the you. Well done, and now we can integrate. We have 1/4 in integral of DU is equal to you. And then we're subtracting the integral of cosine of tou. The integral of cosine as sign. So first of all, we have sign off to you. And since we have a linear term, according to the reverse chain rule, we have to divide by the slope of the linear term, so we're dividing by 2. And of course, we are going to add a constant of integration C. Since we have a difference of indefinite integrals. Now let's replace a U with X2, so we get 1/4. Imparis x square minus. Syme Of 2 X squared. Divided by 2. Plus a constant of integration C and now let's expand. This is going to be equal to 1/4 X2 minus 1/8 of 2 X2 + 1/4 C. Now, since we're working with a constant of integration C, 1/4 C is also C, right? It's just a different constant of integration. We can essentially label our previous constant as C1, and the new constant of integration as C for simplicity. So we're just going to add C at the end. And that will be our final integral for this problem. Thank you for watching.

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.

∫ 𝓍 cos²𝓍² d𝓍

Key Concepts

Trigonometric Identities

Integration Techniques

Definite and Indefinite Integrals

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