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𝘶-substitution with definite integrals

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.D (LO)
,
FUN‑6.D.1 (EK)
,
FUN‑6.D.2 (EK)
Performing u-substitution with definite integrals is very similar to how it's done with indefinite integrals, but with an added step: accounting for the limits of integration. Let's see what this means by finding integral, start subscript, 1, end subscript, squared, start color #7854ab, 2, x, end color #7854ab, start color #e07d10, left parenthesis, end color #e07d10, start color #1fab54, x, squared, plus, 1, end color #1fab54, start color #e07d10, right parenthesis, cubed, end color #e07d10, start color #7854ab, d, x, end color #7854ab.
We notice that start color #7854ab, 2, x, end color #7854ab is the derivative of start color #1fab54, x, squared, plus, 1, end color #1fab54, so u-substitution applies. Let start color #1fab54, u, equals, x, squared, plus, 1, end color #1fab54, then start color #7854ab, d, u, equals, 2, x, d, x, end color #7854ab. Now we substitute:
integral, start subscript, 1, end subscript, squared, start color #7854ab, 2, x, end color #7854ab, start color #e07d10, left parenthesis, end color #e07d10, start color #1fab54, x, squared, plus, 1, end color #1fab54, start color #e07d10, right parenthesis, cubed, end color #e07d10, start color #7854ab, d, x, end color #7854ab, equals, integral, start subscript, 1, end subscript, squared, start color #e07d10, left parenthesis, end color #e07d10, start color #1fab54, u, end color #1fab54, start color #e07d10, right parenthesis, cubed, end color #e07d10, start color #7854ab, d, u, end color #7854ab
Wait a minute! The limits of integration were fitted for x, not for u. Think about this graphically. We wanted the area under the curve start color #11accd, y, equals, 2, x, left parenthesis, x, squared, plus, 1, right parenthesis, cubed, end color #11accd between x, equals, 1 and x, equals, 2.
Function y = 2 x left parenthesis x squared + 1 right parenthesis cube is graphed. The x-axis goes from 0 to 3. The graph is a curve. The curve starts in quadrant 2, moves upward away from the x-axis to (2, 500). The region between the curve and the x-axis, between x = 1 and x = 2, is shaded.
Now that we changed the curve to start color #aa87ff, y, equals, u, cubed, end color #aa87ff, why should the limits stay the same?
Functions y = 2 x left parenthesis x squared + 1 right parenthesis cube and y = u cubed are graphed together. The graph of y = u cubed starts in quadrant 2, moves upward away from the x-axis and ends at about (3, 27).
Both start color #11accd, y, equals, 2, x, left parenthesis, x, squared, plus, 1, right parenthesis, cubed, end color #11accd and start color #aa87ff, y, equals, u, cubed, end color #aa87ff are graphed. You can see the areas under the curves between x, equals, 1 and x, equals, 2 (or u, equals, 1 and u, equals, 2) are very different in size.
Indeed, the limits shouldn't stay the same. To find the new limits, we need to find what values of start color #1fab54, u, end color #1fab54 correspond to start color #1fab54, x, squared, plus, 1, end color #1fab54 for x, equals, start color #ca337c, 1, end color #ca337c and x, equals, start color #ca337c, 2, end color #ca337c:
  • Lower bound: left parenthesis, start color #ca337c, 1, end color #ca337c, right parenthesis, squared, plus, 1, equals, start color #ca337c, 2, end color #ca337c
  • Upper bound: left parenthesis, start color #ca337c, 2, end color #ca337c, right parenthesis, squared, plus, 1, equals, start color #ca337c, 5, end color #ca337c
Now we can correctly perform the u-substitution:
integral, start subscript, start color #ca337c, 1, end color #ca337c, end subscript, start superscript, start color #ca337c, 2, end color #ca337c, end superscript, start color #7854ab, 2, x, end color #7854ab, start color #e07d10, left parenthesis, end color #e07d10, start color #1fab54, x, squared, plus, 1, end color #1fab54, start color #e07d10, right parenthesis, cubed, end color #e07d10, start color #7854ab, d, x, end color #7854ab, equals, integral, start subscript, start color #ca337c, 2, end color #ca337c, end subscript, start superscript, start color #ca337c, 5, end color #ca337c, end superscript, start color #e07d10, left parenthesis, end color #e07d10, start color #1fab54, u, end color #1fab54, start color #e07d10, right parenthesis, cubed, end color #e07d10, start color #7854ab, d, u, end color #7854ab
Functions y = 2 x left parenthesis x squared + 1 right parenthesis cube and y = u cubed are graphed together. The x-axis goes from negative 1 to 6. Each graph moves upward away from the x-axis. The first function ends at (2, 500). The region between the curve and the x-axis between x = 1 and x = 2 is shaded. The second function ends at about (6, 210). The region between the curve and the x-axis, between x = 1 and x = 5, is shaded. The 2 shaded regions look similar in size.
start color #aa87ff, y, equals, u, cubed, end color #aa87ff is graphed with the area from u, equals, 2 to u, equals, 5. Now we can see that the shaded areas look roughly the same size (they are actually exactly the same size, but it's hard to say by just looking).
From here on, we can solve everything according to u:
25u3du=[u44]25=544244=152.25\begin{aligned} \displaystyle\int_{2}^5 u^3\,du&=\left[\dfrac{u^4}{4}\right]_{2}^5 \\\\ &=\dfrac{5^4}{4}-\dfrac{2^4}{4} \\\\ &=152.25 \end{aligned}
Remember: When using u-substitution with definite integrals, we must always account for the limits of integration.
Problem 1
Ella was asked to find integral, start subscript, 1, end subscript, start superscript, 5, end superscript, left parenthesis, 2, x, plus, 1, right parenthesis, left parenthesis, x, squared, plus, x, right parenthesis, cubed, d, x. This is her work:
Step 1: Let u, equals, x, squared, plus, x
Step 2: d, u, equals, left parenthesis, 2, x, plus, 1, right parenthesis, d, x
Step 3:
integral, start subscript, 1, end subscript, start superscript, 5, end superscript, left parenthesis, 2, x, plus, 1, right parenthesis, left parenthesis, x, squared, plus, x, right parenthesis, cubed, d, x, equals, integral, start subscript, 1, end subscript, start superscript, 5, end superscript, u, cubed, d, u
Step 4:
15u3du=[u44]15=544144=156\begin{aligned} \displaystyle\int_1^5 u^3du&=\left[\dfrac{u^4}{4}\right]_1^5 \\\\ &=\dfrac{5^4}{4}-\dfrac{1^4}{4} \\\\ &=156 \end{aligned}
Is Ella's work correct? If not, what is her mistake?
Choose 1 answer:
Choose 1 answer:

Problem 2
integral, start subscript, 1, end subscript, squared, 15, x, squared, left parenthesis, x, cubed, minus, 7, right parenthesis, start superscript, 4, end superscript, d, x, equals, question mark
Choose 1 answer:
Choose 1 answer:

Want more practice? Try this exercise.

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  • cacteye green style avatar for user Kaylagrey
    Could someone please explain how

    ∫sec(u)tan(u)du
    =​sec(u)+C

    Tnx!
    (1 vote)
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  • spunky sam red style avatar for user vanessa hili
    I am having difficulty understanding how to use the differential of du dx. In problem 2 above, how did 15x^2 turn to be 5*3*x^2?
    (1 vote)
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    • female robot grace style avatar for user loumast17
      They are equivalent is the short answer.

      Long answer is, since u was chosen to be x^3 - 7 the derivative is 3x^2, so we want something to substitute that in for. really the important part is x^2. as long as that exists you can "factor out" a 1/3 and then you'll have 1/3 * 3x^2 and have what you need to substitute.

      Luckily, there is 15x^2, and if you factor out a 5 you get the 3x^2 you need, which is where 5*3x^2 came from.

      Does that help? if not I can maybe try walking through it differently than the problem.
      (2 votes)
  • blobby green style avatar for user orel7403
    what if the bounds get flipped after changing them as in you would be integrating from a to b when b<a can you still leave the u sub or do you have to swap them
    (1 vote)
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    • sneak peak green style avatar for user Jeff
      If the bounds become inverted (b<a) due to a u-sub, it is typically best switch them back. It is OK to switch the bounds as long as you add a negative out front of the integral to make up for it.

      If you don't fix the "backwards bounds" you will still end up with the same answer in the end. However, it's good practice to swap them and add the negative.
      (1 vote)
  • leaf green style avatar for user 石乐志大师
    Why does the definite integral of u^3 from 2 to 5 has the same area as the definite integral of 2x(x^2+1)^3 from 1 to 2?
    (1 vote)
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  • blobby green style avatar for user gunank312
    i tried to plot graph of 2x(x^2 + 1)^3 and (x^2 + 1)^3 , but why am i not getting the same graphs as depicted? we are trying to find area under curve given the bounds only right?
    (1 vote)
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  • sneak peak green style avatar for user escapingperil
    I am working on a question like this and having trouble. It is asking me to find the area between the curve y=xsqrt(4-x^2) and the x-axis over the interval [-2, 2].

    I am mostly confused on how not to get zero as an answer. Should I break it into multiple parts?
    (1 vote)
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  • aqualine ultimate style avatar for user caiyresl
    Can you substitute the original expression with x for u and then use the same boundaries? In the example, would (x^2+1)^4/4 for [1,2] work?
    (1 vote)
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    • blobby green style avatar for user Misha Taylor
      Exactly. It's easier that way. The Limits might get too big. With Polynomials and cubic and quintic equations more shortcuts are ready, since (a+b)^5=a^5+b^5+5ab(a+b)^3+5ab(a+b), and One can use it more if the have a quintic nested cubic or vice versa, when doing crazy stuff with diffEQ and regression to a smaller diffEQ that's easier. If You're dealing with a+b, rather have a^5+b^5 or (a+b)^5 be ready. If a^5+b^5=x, and a+b=y, and ab=c. I'd think of something like x=y^5-5cy^3+5cy; You put in the limits, it isn't as ready for running parallel processes. Running parallel processes in Math is the Best Part of ADHD and Autism, and it finds out more information when connecting the dots gets connected. If You don't run enough parallel processes, you might not find the quickest solution. For Some Reason, 42*6, can be rewritten as 36*7, faster than multiplying 6*7, because parallel processes were "strong as a carbon atom". Multiplying 512*84, can be quickly rewritten as 42000+24*42, which can be quickly done be rewritten it again as 12*84 which->1200-12*16->1200-(196-16), and so on. Verify that by another parallel process like 4096*12 which is 480000+1200-48 which is probably 481152. If That's wrong, just create another parallel process. Verify it by thinking does 4*2 end in 4. Nope! So 2048*21=42000+1000-((1*48-2*21=>6)+2=>8)->43008. Or 1050-42->8+43000->43008. Run More Threads and so on.
      (1 vote)