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
$\int f\left(u\right)du}.$
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest.
Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.?
How this robot is carried to required site of body cell.?
what will be the carrier material and how can be detected that correct delivery of drug is done
Rafiq
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Privacy Information Security Software Version 1.1a
Good
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?