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
$\int f\left(u\right)du}.$
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
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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?