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SmartyPants

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- TL;DR Summary
- Used Math DF's integral calculator to check a solution I got, and I think it might be making a big mistake in its calculations...

In calculating the integral ##\int{\ln\left(x\right)\,\sin\left(x\right)\,\cos\left(2\,x\right)}{\;\mathrm{d}x}##, I used a few online integral calculators to check my answer. According to one calculator, I got the correct antiderivative, but according to another (Math DF Integral Calculator), I did not.

As you know, the antiderivative of a function can often take more than one form (and sometimes several forms) depending on how one goes about solving the integral. For instance, we know that ##\int tan~x~dx## = both ##ln\left | sec~x \right |+C## and ##-ln\left | cos~x \right |+C##. And while it may be straightforward to show that these two solutions are equivalent, it remains to be seen as to whether the two different solutions these online integral calculators came up with for ##\int{\ln\left(x\right)\,\sin\left(x\right)\,\cos\left(2\,x\right)}{\;\mathrm{d}x}## are in fact equivalent.

But for the moment, I'm more concerned with whether or not Math DF's integral calculator got it wrong than I am concerned with showing that two different solutions to ##\int{\ln\left(x\right)\,\sin\left(x\right)\,\cos\left(2\,x\right)}{\;\mathrm{d}x}## are equivalent. Math DF's solution is as follows (it'll be easier for me to just post screen shots than it will to put it all in LaTex code here, as some of the LateX code at Math DF can be copied, but much of it can't be copied):

The calculator chooses to go with Integration by Parts to start, and right away, in breaking the integrand up into a ##u## and a ##dv##, it doesn't account for the entire integrand. Specifically, it chooses ##u=sin~x## and ##dv=cos~2x~dx##, and leaves the ##ln~x## out completely. These IBP calculations would make sense for ##\int sin~x~cos~2x~dx##, but not for ##\int{\ln\left(x\right)\,\sin\left(x\right)\,\cos\left(2\,x\right)}{\;\mathrm{d}x}##...unless am I overlooking something.

As for this second part of their solution, the calculations I see here seem to make good sense so long as the calculations that got them to this point (performing IBP twice) are correct. But again, I can't be sure that this calculation is correct because I don't know that they got their integration by parts correct in the first place...and on the surface, it appears that they got the IBP calculations wrong.

In summary, are their IBP calculations correct and am I just overlooking something simple? Or did Math DF's integral calculator make a blatant error in leaving the ##ln~x## out of their IBP calculations?

Thanks in advance,

Eric

As you know, the antiderivative of a function can often take more than one form (and sometimes several forms) depending on how one goes about solving the integral. For instance, we know that ##\int tan~x~dx## = both ##ln\left | sec~x \right |+C## and ##-ln\left | cos~x \right |+C##. And while it may be straightforward to show that these two solutions are equivalent, it remains to be seen as to whether the two different solutions these online integral calculators came up with for ##\int{\ln\left(x\right)\,\sin\left(x\right)\,\cos\left(2\,x\right)}{\;\mathrm{d}x}## are in fact equivalent.

But for the moment, I'm more concerned with whether or not Math DF's integral calculator got it wrong than I am concerned with showing that two different solutions to ##\int{\ln\left(x\right)\,\sin\left(x\right)\,\cos\left(2\,x\right)}{\;\mathrm{d}x}## are equivalent. Math DF's solution is as follows (it'll be easier for me to just post screen shots than it will to put it all in LaTex code here, as some of the LateX code at Math DF can be copied, but much of it can't be copied):

The calculator chooses to go with Integration by Parts to start, and right away, in breaking the integrand up into a ##u## and a ##dv##, it doesn't account for the entire integrand. Specifically, it chooses ##u=sin~x## and ##dv=cos~2x~dx##, and leaves the ##ln~x## out completely. These IBP calculations would make sense for ##\int sin~x~cos~2x~dx##, but not for ##\int{\ln\left(x\right)\,\sin\left(x\right)\,\cos\left(2\,x\right)}{\;\mathrm{d}x}##...unless am I overlooking something.

As for this second part of their solution, the calculations I see here seem to make good sense so long as the calculations that got them to this point (performing IBP twice) are correct. But again, I can't be sure that this calculation is correct because I don't know that they got their integration by parts correct in the first place...and on the surface, it appears that they got the IBP calculations wrong.

In summary, are their IBP calculations correct and am I just overlooking something simple? Or did Math DF's integral calculator make a blatant error in leaving the ##ln~x## out of their IBP calculations?

Thanks in advance,

Eric

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