Hello, everyone. Tom from everystepcalculus.com and everystepphyics.com. A problem in calculus.
Integration by Parts. Let’s do it. Sine. So let’s do it. Scroll down to integration by parts. I’m already there to save time. It’s called Integrate transcendentals. And we’re going to choose number 3, Sine. And we’re going to enter our function. You have to press Alpha before you enter anything into these entry lines, here. Alpha x times sine of 3 times, make sure you add the times sign between in math, it’s a good practice. Not only for my programs but for any program or anything in the calculator. You have to tell the calculator and me what you want. Not just do what professors do on the blackboard. which is put 3x. I always show you’ve entered, you can change it if you want. I say it’s okay. And dv then is the sine of 3x and the integral of sine of 3x is this minus cosine 3x divided by 3 and that’s v. Should be a plus c. I don’t know integration by parts. There is a plus 3 c after this but the geniuses in Calculus just kind of throw that off and I don’t why. And x is the u and the derivative of x is one. So now the formula v times u minus the integral of v times du dx. A lot of books have u times v. I like to put it v times u because here we have v times du so we have the same. So we add, you add that v minus cosine of 3x divided by 3 times x. And the integral of v which is v, here’s v again and then du is one. So you just add the things to the formula, which we’re doing here. And this turns out to be this minus the cosine of 3x divided by 3 and then we have the same thing in here. Anytime you have a constant in the integral, you take it out. So here’s the, take the one third out of the integral and then integrate minus cosine of 3x. And minus x times cosine of 3 x is equal to minus one third minus the sine of 3x divided by 3. Now we bring the third out here again and that’s where you get the one ninth. So the answer is this minus x cosine of 3 divided by 3 and here’s one ninth times sine of 3x plus C. Pretty neat, huh? Have a good one.
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