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Course: Calculus 2 > Unit 2

Lesson 1: Integrating with u-substitution

𝘶-substitution with definite integrals

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

\int_{1}^{2} 2 x (x^{2} + 1)^{3} d x

We notice that

2 x

is the derivative of

x^{2} + 1

, so

u

-substitution applies. Let

u = x^{2} + 1

, then

d u = 2 x d x

. Now we substitute:

\int_{1}^{2} 2 x (x^{2} + 1)^{3} d x = \int_{1}^{2} (u)^{3} d u

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

y = 2 x (x^{2} + 1)^{3}

between

x = 1

and

x = 2

Now that we changed the curve to

y = u^{3}

, why should the limits stay the same?

Indeed, the limits shouldn't stay the same. To find the new limits, we need to find what values of

u

correspond to

x^{2} + 1

for

x = 1

and

x = 2

Lower bound: $(1)^{2} + 1 = 2$ ‍
Upper bound: $(2)^{2} + 1 = 5$ ‍

Now we can correctly perform the

u

-substitution:

\int_{1}^{2} 2 x (x^{2} + 1)^{3} d x = \int_{2}^{5} (u)^{3} d u

y = u^{3}

is graphed with the area from

u = 2

u = 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

\begin{aligned} \int_{2}^{5} u^{3} d u & = {[\frac{u^{4}}{4}]}_{2}^{5} \\ = \frac{5^{4}}{4} - \frac{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

\int_{1}^{5} (2 x + 1) (x^{2} + x)^{3} d x

. This is her work:

Step 1: Let

u = x^{2} + x

Step 2:

d u = (2 x + 1) d x

Step 3:

\int_{1}^{5} (2 x + 1) (x^{2} + x)^{3} d x = \int_{1}^{5} u^{3} d u

Step 4:

\begin{aligned} \int_{1}^{5} u^{3} d u & = {[\frac{u^{4}}{4}]}_{1}^{5} \\ = \frac{5^{4}}{4} - \frac{1^{4}}{4} \\ = 156 \end{aligned}

Is Ella's work correct? If not, what is her mistake?

Problem 2

\int_{1}^{2} 15 x^{2} (x^{3} - 7)^{4} d x = ?

Want more practice? Try this exercise.

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