48. Integral of sec³x Use integration by parts to show that:

Question

∫ sec³x dx = (1/2) secx tanx + (1/2) ∫ secx dx

Accepted Answer

Hello. In this video, we are going to be using integration by parts to find the following integral. We are given the integral of co-seeking cubed YDY. Now, in order to use integration by parts, the first thing we're going to do is we're going to rewrite the given function as a product of cosequent functions. Let's go ahead and rewrite the integral to be the integral of cosequant of Y multiplied by cosequent squared of Y D Y. Then by the integration by parts method, we are going to divine you to be co-seeking of Y. And we're going to define our DV function to be cosequent square to Y. Now, by taking the derivative of the U statement, we will end up with DU is equal to negative cosequant of Y. Cottingent of wine. And by integrating our DV function, V is going to equal to negative cotangent of Y. And so once we replace or apply our integration by parts, we are going to be left with U multiplied by V, which is going to be negative cosequent of Y, multiplied by cotangent of Y minus the integral of V multiplied by DU, which is going to be negative cotangent of Y, multiplied by negative cosequent of Y cotangent of Y, and that is going to simplify to be cos seek into Y. Multiplied by cotangent squared of y. Now, there's a bit of simplification that we can do here. If we take a look at the given integral, we can rewrite the integral by using the Pythagorean identity. If we were to rewrite cotangent in terms of cosequent, this integral can be replaced with cosequent of Y multiplied by the quantity cosequent square of Y minus 1. Then if we were to distribute cosequant into the quantity, we will have the integral of cosequant cubed of Y minus cosequant of Y. Then what we can go ahead and do is apply or distribute the integral symbol amongst the different terms and have a difference of integrals. So we will have the integral of cosequent cubed of Y minus cosequant of Y. Or the integral of Kosik in a Y. And keep in mind that we still have to distribute this negative sign afterward, so once we distribute the negative, we will have negative integral of cosequent cubed of Y plus the integral of cosequant of Y. So by applying these, by applying this change, we know that so far the integral of cosequent cube to Y. D Y is equal to negative cosequant of y. Multiplied by the cotangent of Y. Minus the integral of cosequant cubed of y. Plus the integral of cosequant of Y. Now, here, there is a tricky simplification that we can apply here. Notice that we have co-seeking cubed of Y in the integration by parts. If we try to do integration by parts again, we're going to end up in this loop where we keep getting the integral of the cos cubed to Y. Now we know that the original integral is equal to this current expansion. So what we can go ahead and do is we can add the integrals of co-seeking cub to Y with each other. So we are going to add the integral of coseeking cub to Y to our original co-seeking cub to Y. That is going to leave us with 2 multiplied by the integral of co-seeking q to Y. And that is going to equal to the negative cosi in a Y. Cotingent of wine. Plus the integral. Of Kosikin of wine. Then if we divide by the coefficient of 2 on the left hand side of the equation, we will be left with the integral of Cosik and cubed to Y, and that is equal to negative 1/2 multiplied by cok into Y cotangent to Y. Plus 1/2 multiplied by the integral of cosequent of Y. Now, what we need to do in this case is we need to simplify the integral of cose and of Y. By the identity, the integral of Kosikin of y. It is equal To the negative natural logarithm. Of Kosikintawa. Plus cotangent of Y. So, once we apply this identity, we can further simplify the integration by parts to be negative 12 multiplied by cos seek in a Y, cotangent a Y. Minus 1/2 multiplied by the natural logarithm of coeek into Y. Plus cotangent of y. And finally, what we can go ahead and do here is we can factor out a -12, and once we do that, we will have -12 multiplied by the quantity cossein of Y. Kotangina wine Plus the natural logarithm of cosequence of y. Plus cotangent of wine. And because we were working with an indefinite integral in the beginning, we will add a general constant C. And this is going to be the solution to the original integral by using integration by parts. So I hope this video helps you in understanding how to approach this type of problem, and we will go ahead and see you all in the next video.

48. Integral of sec³x Use integration by parts to show that:
∫ sec³x dx = (1/2) secx tanx + (1/2) ∫ secx dx

Key Concepts

Integration by Parts

Trigonometric Identities

Integral of sec x

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