Video: Finding the Integral of a Function Using Integration by Parts

Integrate ∫ ln π‘₯ dπ‘₯ by parts using 𝑒 = ln π‘₯ and d𝑣 = dπ‘₯.

01:35

Video Transcript

Integrate the natural log of π‘₯ dπ‘₯ by parts using 𝑒 equals the natural log of π‘₯ and d𝑣 equals dπ‘₯.

Well, this is great. We’ve been told to use integration by parts. And we’ve also been told to let 𝑒 be equal to the natural log of π‘₯ and d𝑣 be equal to dπ‘₯. So we recall the formula for integration by parts. The integral of 𝑒 times d𝑣 by dπ‘₯ is equal to 𝑒𝑣 minus the integral of 𝑣 times d𝑒 by dπ‘₯. Now we can rewrite d𝑣 equals dπ‘₯ slightly. And we can say that if d𝑣 is equal to dπ‘₯, then d𝑣 by dπ‘₯ must be equal to one. Our job is going to be to find the missing parts of the formula. d𝑒 by dπ‘₯ is fairly straightforward. The derivative of the natural log of π‘₯ is one over π‘₯. And if we integrate both sides of the equation d𝑣 equals dπ‘₯, we obtain 𝑣 as being equal to π‘₯. Substituting everything into the formula, and we get π‘₯ times the natural log of π‘₯ minus the integral of π‘₯ times one over π‘₯ dπ‘₯.

Well, this integral simplifies really nicely. We’re actually going to be integrating one with respect to π‘₯. The integral of one is, of course, simply π‘₯. And of course, since this is an indefinite integral, we must remember to add that constant of integration 𝑐. So we obtain the integral of the natural log of π‘₯ dπ‘₯ as being π‘₯ times the natural log of π‘₯ minus π‘₯ plus 𝑐. And integration by parts was really effective for integrating the natural log of π‘₯. Because the derivative of the natural log of π‘₯ is much simpler than the original function the natural log of π‘₯.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.