Evaluate the integrals in Exercises 1–24 using integration by ...

Question

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ 4x sec²(2x) dx

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Compute the integral of x multiplied by c square of 3 x dx using integration by parts. OK, so it appears for this particular problem we're asked to solve for the integral that is provided to us by using integration by parts method. So now that we know what we're ultimately trying to solve for, we must first recall and use, or rather we need to first recall how to use integration by parts. So, we need to identify parts of the integram to assign as U and DV. Next, we need to compute the corresponding derivatives and integrals for each part. Then we will need to at that point, apply the integration by parts formula to the integral. Afterwards, we will need to integrate the resulting simpler integral that we produced. And finally, at the very end, we will need to substitute back in our variables in order to obtain the final solution. So with that said, let us choose our parts. So let us say that you. Will be equal to x. And let us state that DV will be equal to drum roll, C can't square of 3x, so be equal to c square. Of 3 X. Fantastic. So once again, we just chose our parts to use for our integration by parts method. So now we need to find the derivative of U is equal to X. So once again we're finding the derivative of U is equal to X to get that DU by DX is equal to 1. So, that means we need to multiply both sides by DX to get DU is equal to DX. So now, we need to find the integral of DV is equal to c 2 of 3 XDX. OK. So that means when you take the integral of DV is equal to the integral of c 23 XDX, which will mean that V is equal to 1/3 multiplied by tangent of 3 X. So, at this point, we now need to We need to apply our integration by parts formula, which, as we should recall, states that the integral of UDV is equal to u multiplied by V minus the integral of VDU. So, with that in mind, we need to take the integral of x multiplied by c square of 3 xDX, which will be equal to x multiplied by 1/3 multiplied by tangent of 3 x minus the integral of 1 divided by 3 or 1/3. Multiplied by tangent of 3 X DX fantastic, which will be equal to. X divided by 3 multiplied by tangent of 3 X minus 1/3 multiplied by the integral of tangent of 3 XDX. OK. Which will be equal to X divided by 3. Multiplied by tangent. Of 3 X, so multiplied by tangent. Uh 3 X. So, tangent of 3 X. Fantastic. So 3 X. Fantastic. Here we go. And then we need to subtract 1 divided by 3, multiplied by parentheses of 1/3, multiplied by the natural log of the absolute value of cosine of. 3 X. Plus C where C denotes some constant variable, which will mean that this will be equal to x divided by 3 multiplied by tangent of 3 x plus 1 divided by 9 multiplied by the natural log of the absolute value of cosine of 3 x plus c. So that will mean that our final answer will be the integral of X multiplied by C2. Of 3 XDX 3 XDX will be equal to 1 divided by 3 multiplied by x multiplied by tangent of 3 x plus 1 divided by 9 multiplied by the natural log of the absolute value of cosine of 3 x plus C. And that's it. That is our final answer. Hooray, we did it. So going back to look at the prom itself to recap what we've learned, so that will mean our final answer once again. Without a doubt will be 1/3 multiplied by x multiplied by tangent of 3 x minus 1/9 multiplied by the natural log of the absolute value of cosine of 3 x plus c, and that's it. That is our correct and final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Evaluate the integrals in Exercises 1–24 using integration by parts.∫ 4x sec²(2x) dx

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Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ 4x sec²(2x) dx