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U-Substitution

This original textbook reading covers part of the AP topic. Follow the AP course outline for complete topic instruction and practice.

Complex integrals

You might recall how complex taking some derivatives got. What if we wanted to take the integral of a derivative that used the chain rule? Often times it is not intuitive. Finding integrals can often be much more difficult than finding derivatives. The formal method of solving integrals that are derived from using the chain rule is called u-substitution.

We set u equal to the inside function, and use the three-step substitution method discussed in Section 8.4. This requires a bit of practice and intuition, so this section will have plenty of that.

Example Problem 9.1.1

Evaluate the indefinite integral

$\int 3 x^{2} \cos (x^{3} + \ln (x) + 3) + \frac{\cos (x^{3} + \ln (x) + 3)}{x} d x$

This may seem very complex, but notice that we can factor out the cosine term out of both parts of the integrand.

$= \int (3 x^{2} + \frac{1}{x}) \cos (x^{3} + \ln (x) + 3) d x$

Now let’s substitute:

$u = x^{3} + \ln (x) + 3$
$d u = (3 x^{2} + \frac{1}{x}) d x$

Notice that the integral is now written in such a way that these substitutions just fall right into place.

$= \int \cos (x^{3} + \ln (x) + 3) \cdot (3 x^{2} + \frac{1}{x}) d x$

$= \int \cos u d u$
$= \sin u + C$

$= s i n (x^{3} + l n (x) + 3) + C$

Example Problem 9.1.2

Evaluate the definite integral $\int_{2}^{5} \frac{2 x^{2} + x + 2}{x - 1} d x$

When choosing where to substitute, it is often the instinct to substitute the most complex part of the integral. However, for fractional integrals, it is better to substitute for the denominator, because the numerator can be split up into smaller integrals.

$u = x - 1$
$d u = d x$ (derivative of a constant is 0)
Since x: $2 \to 5$ , then u: $1 \to 4$

Substituting these gives us the integral

$\int_{1}^{4} \frac{2 x^{2} + x + 2}{u} d u$

What can we do about the numerator though? Well, the great thing about substitution is, just as we can set u as a function of x, we can also set x as a function of u. Rearranging $u = x - 1$ into $x = u + 1$

$\int_{1}^{4} \frac{2 (u + 1)^{2} + (u + 1) + 2}{u} d u$

$= \int_{1}^{4} \frac{2 (u^{2} + 2 u + 1) + u + 3}{u} d u$

$= \int_{1}^{4} \frac{2 u^{2} + 4 u + 2 + u + 3}{u} d u$
$= \int_{1}^{4} \frac{2 u^{2} + 5 u + 5}{u} d u$

Now we can split up this integral and integrate each fraction individually.

$= \int_{1}^{4} \frac{2 u^{2}}{u} d u + \int_{1}^{4} \frac{5 u}{u} d u + \int_{1}^{4} \frac{5}{u} d u$

$= \int_{1}^{4} u^{2} + \int_{1}^{4} 5 u + 5 \ln (u)$

${[u^{2}]}_{1}^{4} + {[5 u]}_{1}^{4} + {[5 \ln u]}_{1}^{4}$

$= (4^{2} - 1^{2}) + 5 (4 - 1) + 5 (\ln (4) - \ln (1))$

$= 15 + 15 - 5 \ln (4)$ $= 5 (6 - l n (4))$

Glossary and deeper reading · 3 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
Read more: 2.2 · Derivative →
Integral
Integration can mean accumulating a function over an interval or finding an antiderivative. A definite integral gives a number, while an indefinite integral represents a family of antiderivatives. For example, integrating 2x indefinitely gives x²+C, and integrating it from 0 to 2 gives 4. The Fundamental Theorem of Calculus connects these two operations under its hypotheses.
Read more: 6.7 · Integral →
Substitution for integration
Substitution rewrites an integral using a new variable whose differential accounts for a derivative factor. It reverses the chain rule for suitable compositions. For example, with u=x²+1 and du=2x dx, the integral of 2x(x²+1)³ becomes the integral of u³. For a definite integral, transform the bounds consistently or return to the original variable before evaluating them.
Read more: 6.9 · Substitution for integration →

Pop Quiz 9.1.1 U-Substitution

Derivative

Integral

Substitution for integration

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