Using substitution to evaluate a definite integral
Use substitution to evaluate
${\int}_{0}^{1}{x}^{2}{\left(1+2{x}^{3}\right)}^{5}}dx.$
Let
$u=1+2{x}^{3},$ so
$du=6{x}^{2}dx.$ Since the original function includes one factor of
x^{2} and
$du=6{x}^{2}dx,$ multiply both sides of the
du equation by
$1\text{/}6.$ Then,
Substitution may be only one of the techniques needed to evaluate a definite integral. All of the properties and rules of integration apply independently, and trigonometric functions may need to be rewritten using a trigonometric identity before we can apply substitution. Also, we have the option of replacing the original expression for
u after we find the antiderivative, which means that we do not have to change the limits of integration. These two approaches are shown in
[link] .
Using substitution to evaluate a trigonometric integral
Use substitution to evaluate
${\int}_{0}^{\pi \text{/}2}{\text{cos}}^{2}\theta \phantom{\rule{0.2em}{0ex}}d\theta}.$
Let us first use a trigonometric identity to rewrite the integral. The trig identity
${\text{cos}}^{2}\theta =\frac{1+\text{cos}\phantom{\rule{0.1em}{0ex}}2\theta}{2}$ allows us to rewrite the integral as
We can evaluate the first integral as it is, but we need to make a substitution to evaluate the second integral. Let
$u=2\theta .$ Then,
$du=2d\theta ,$ or
$\frac{1}{2}du=d\theta .$ Also, when
$\theta =0,u=0,$ and when
$\theta =\pi \text{/}2,u=\pi .$ Expressing the second integral in terms of
u , we have
Substitution is a technique that simplifies the integration of functions that are the result of a chain-rule derivative. The term ‘substitution’ refers to changing variables or substituting the variable
u and
du for appropriate expressions in the integrand.
When using substitution for a definite integral, we also have to change the limits of integration.
Key equations
Substitution with Indefinite Integrals $\int f\left[g\left(x\right)\right]{g}^{\prime}\text{(}x)dx={\displaystyle \int f\left(u\right)du}}=F\left(u\right)+C=F\left(g\left(x\right)\right)+C$
Substitution with Definite Integrals $\int}_{a}^{b}f\left(g\left(x\right)\right)g\text{'}\left(x\right)dx={\displaystyle {\int}_{g\left(a\right)}^{g\left(b\right)}f\left(u\right)du$
Why is
u -substitution referred to as
change of variable ?
2. If
$f=g\circ h,$ when reversing the chain rule,
$\frac{d}{dx}(g\circ h)(x)={g}^{\prime}\text{(}h(x)){h}^{\prime}\text{(}x),$ should you take
$u=g\left(x\right)$ or
$u=h\left(x\right)?$
In the following exercises, verify each identity using differentiation. Then, using the indicated
u -substitution, identify
f such that the integral takes the form
$\int f\left(u\right)du}.$
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Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?