Evaluate the integrals in Exercises 31–56. Some integrals do n...

Question

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ x² sin(x³) dx

Accepted Answer

Welcome back, everyone. Evaluate the integral X to the power of 4 multiplied by cosine of X to the power of 5DX. Let's approach this problem using U substitution, and let's suppose that U is equal to X to the power of 5. What we want to do now is differentiate you with respect to X. That's the derivative of X the power of 5, which is equal to 5x the power of 4. Notice that within the integrant we have X the power of 4DX. So let's go ahead and rearrange this expression. We can divide both sides by 5 and multiply both sides by DX, right? To get 1/5 multiplied by DU. Equals x to the power of 4D X. So now this allows us to rewrite the integral in terms of UNDU. We have cosine of you. Multiplied by X to the power of 4 multiplied by D X is 1/5 DU. So we can write DU and we can fax route 1/5. You know this is an elementary integral. We have 1/5, and the integral of cosine of UDU is sign of you. We're going to add the constant of integration C for an indefinite integral and replace you with X to the power of 5, which gives us 1/5 of X to the power of 5 plus C. That's our final answer and thank you for watching.

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.∫ x² sin(x³) dx

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Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x² sin(x³) dx