Why Does No One Teach This? 😭

Added: 2026-05-25

[00:00:00]Today, I want to show you a calculus trick that can genuinely change the way you do integration problems. This trick is often associated with the book Elementary Analysis, The Theory of Calculus by Kenneth Ross, and many calculus lovers call it a life-changing integration trick because once you see it, you might start using it everywhere.

[00:00:23]Now, this is the usual way we perform integration by parts, right?

[00:00:28]Suppose we want to integrate log of x + 3.

[00:00:32]The standard approach is to rewrite it as 1 multiplied by log of x + 3, and then apply integration by parts. Most students take the antiderivative of 1 as simply x, so they get x multiplied by log of x + 3 minus integral of x divided by x + 3. And now, the remaining integral starts looking messy. This is where we now have to perform more algebraic manipulations, right? But, here comes the interesting observation.

[00:01:06]Antiderivatives are never unique because we can always add a constant to them.

[00:01:11]For example, if x squared divided by 2 is an antiderivative of x, then x squared divided by 2 + 5 is also an antiderivative.

[00:01:23]Both are equally correct because constants disappear during differentiation.

[00:01:28]And this small thing is exactly what makes the trick so powerful.

[00:01:32]So now, let us apply the Ross trick.

[00:01:35]Instead of choosing just x as the antiderivative of 1, remember that we are allowed to add any constant we want.

[00:01:43]So instead, let us choose x + 3.

[00:01:47]Now, apply integration by parts again.

[00:01:51]Suddenly, the integrand during integration by parts disappears, and it becomes x + 3 divided by X plus three, which is simply one. Now, integral of one is simply X. Therefore, the final answer becomes X plus three multiplied by log of X plus three minus X plus constant.

[00:02:13]Wow, that is indeed super cool.

[00:02:16]Now, let us move to another example where we want to integrate X multiplied by tan inverse of X.

[00:02:24]Most students begin normally by choosing the X as first function and tan inverse as the second function.

[00:02:31]Then, integration by parts gives X squared divided by two multiplied by tangent inverse of X minus integral of X squared divided by two multiplied by derivative of tangent inverse, which is one divided by X squared plus one.

[00:02:49]And now, the remaining fraction looks unpleasant because simplifying it takes extra work.

[00:02:55]But again, instead of choosing only X squared divided by two, let us cleverly choose X squared plus one whole divided by two.

[00:03:06]We simply added one to this X squared.

[00:03:09]Now, look carefully at what happens next.

[00:03:13]During integration by parts, the numerator X squared plus one cancels perfectly with the denominator X squared plus one.

[00:03:22]So, the horrible looking integral suddenly reduces to the integral of one divided by two.

[00:03:28]That means the answer immediately becomes X squared plus one whole divided by two multiplied by tan inverse of X minus X divided by two plus constant.

[00:03:41]Compare this with the longer traditional solution.

[00:03:44]Once you see this pattern, many difficult integration problems start looking much simpler.

[00:03:51]Now, let us try one slightly more advanced example.

[00:03:54]Suppose we want to integrate tangent inverse of square root of x plus four.

[00:04:00]Again, we rewrite it as one multiplied by tangent inverse of square root of x plus four.

[00:04:07]Its integration by parts will give us this result, right?

[00:04:11]Now, when we differentiate tangent inverse of square root of x plus four using the chain rule, the denominator eventually becomes one plus square root of x plus four whole squared, which simplifies into x plus five. This times one over two times square root of x plus four.

[00:04:32]Now, this observation is the key idea.

[00:04:35]Since we know x plus five will appear in the denominator later, we cleverly choose x plus five as the antiderivative instead of just x.

[00:04:46]Now, after applying integration by parts, the denominator x plus five cancels perfectly with the numerator x plus five.

[00:04:55]So, the complicated looking integral collapses into a very simple integral.

[00:05:00]This is straightforward because the integral of one divided by two times the square root of x plus four is simply the square root of x plus four.

[00:05:09]So, the final answer becomes x plus five multiplied by tan inverse of square root of x plus four minus square root of x plus four plus constant. This is simply out of the world.

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[00:05:36]So good.

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