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

Integrate ∫ ln 𝑥 d𝑥 by parts using 𝑢 = ln 𝑥 and d𝑣 = d𝑥.

01:35

Video Transcript

Integrate the natural log of 𝑥 d𝑥 by parts using 𝑢 equals the natural log of 𝑥 and d𝑣 equals d𝑥.

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

Well, this integral simplifies really nicely. We’re actually going to be integrating one with respect to 𝑥. The integral of one is, of course, simply 𝑥. 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 𝑥 d𝑥 as being 𝑥 times the natural log of 𝑥 minus 𝑥 plus 𝑐. And integration by parts was really effective for integrating the natural log of 𝑥. Because the derivative of the natural log of 𝑥 is much simpler than the original function the natural log of 𝑥.

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