Chain rule integration plus#

Sine of x to the fifth power over five plus c and we are done. So we're gonna get sine to the third, sine of x to the third over three minus sine of x to the fifth power. Then we just do the reverse substitution and then that gets us to be, instead of u we want to put a sine of x there. This is pretty straightįorward, now this is going to be u to the third over three minus u to the fifth over five plus c. We can rewrite that as, u squared minus u to the fourth times du. X we're saying sine is the same thing as u so The anti-derivative of, it's over the whole stretch. If we say that u is equal to sine of x, then du is going to be equal to cosine of x dx, and that works out quite well because we have the du right over here. The whole reason why we did this little algebraic manipulation. Minus sine to the fourth x, but I have the derivative The first rule to know is that integrals and derivatives are opposites Sometimes we can work out an integral, because we know a matching derivative. It is often used to find the area underneath the graph of a function and the x-axis. Now this is starting to look interesting, cause I have sine squared x Integration can be used to find areas, volumes, central points and many useful things. Indefinite integral of, Sine squared x times one is going to be sine squared x and then sine squared x times negative sine squared x is negative sine to the fourth. Sine squared times one minus sine squared and I am left with. Identity, is the same thing as one minus sine squared x and then we have cosine x, times cosine x dx. Please read the guidance notes here, where you will find useful information for running these types of activities with your students. The Pythagorean Identity to convert this into This type of activity is known as Practice. And this could be rewritten as, sine squared x, then I'm going to use Power times that thing to the first power, dx. Sometime to the third power, something to the second So this can be rewritten as sine squared x times. X, what do I mean by that? Let me just rewrite this, The Pythagorean Identity with the remaining cosine squared Separate out one of the cosine's x and then use That, if you have an odd exponent like this, is to This expression inside, so that you can use u-substitution. What you do is you try to algebraically engineer We can use this method to find an integral value when it is set up in the special form. That this one does have an odd exponent, theĬosine has an odd exponent. In calculus, the integration by substitution method is also known as the Reverse Chain Rule or U-Substitution Method.

If one of these has an odd exponent and you see Is equal to sine of x, 'but I can't do that over here.' So the general ideas here, 'Likewise, if this was justĪ cosine of x not a cosine 'to the third of x, I could

'Well that's going to be the negative of 'the derivative of cosine of x. You look at it you're like, 'Oh wow, if this was just a sine of x, 'not a sine squared of x. Video and see if you can work it through on your own. We can take the indefinite integral of sine squared xĬosine to the third x dx. Therefore, the answer is -eᵡcot(x) + C, where C is an arbitrary real number. When we plug this into the expression ∫ eᵡ / sin²(x) dx - ∫ eᵡcot(x) dx, we get Using integration by parts on the expression ∫ eᵡ / sin²(x) dx yields We will first focus on the first of these integrals. This may be split up into two integrals as ∫ eᵡ / sin²(x) dx - ∫ eᵡcot(x) dx. To solve the integral, we will first rewrite the sine and cosine terms as follows:

Chain rule integration