Question

Answer the question, concerning the use of substitution in integration.

If we use

u equals x squaredu=x2

as a substitution to find

Integral from nothing to nothing x e Superscript x squared Baseline dx comma∫xex2 dx,

then which of the following would be a correct result?

Answer #1

Prove that for positive integers,
Integral from nothing to nothing tangent Superscript n Baseline
x dx equals StartFraction tangent Superscript n minus 1 Baseline x
Over n minus 1 EndFraction minus Integral from nothing to nothing
tangent Superscript n minus 2 Baseline x dx comma n not equals
1∫tannx dx=tann−1xn−1−∫tann−2x dx, n≠1.
Use the formula to evaluate
Integral from 0 to StartFraction pi Over 3 EndFraction 4 tangent
Superscript 5 Baseline x dx∫0π34tan5x dx.

To compute the integral ∫cos3(x)esin(x)dx, we should use first
the substitution u= __________ to obtain the integral ∫F(u)du,
where F(u)= __________ To compute the integral ∫F(u)du, we use the
method of integration by parts with
f(u)= __________ and g′(u)= __________ to get
∫F(u)du=G(u)−∫H(u)du , where G(u)= __________ and H(u)= __________
Now, to compute the integral ∫H(u)du, we need to use the method of
integration by parts a second time with f(u)= __________ and g′(u)=
__________ to get
∫H(u)du= __________ +C...

7.1 To solve ∫ ?? / (√7−?^(2)) by trig
substitution, we should set x = which of the following ?
x = sin θ x = 7 tan θ x = 7 sin θ
x = √7 sin θ x = 72 sin θ x = 7 sin2 θ
7.2 To use Integration by Parts with ∫e^(2x) x^(2) dx , we should
choose which ?
u = x and dv = ex dx u = e2x and...

Evaluate the integral using integration by parts with the
indicated choices of u and dv. (Use C
for the constant of integration.)
xe5xdx; u
= x, dv =
e5xdx
2. Evaluate the integral. (Use C for the constant of
integration.)
(x2 + 10x) cos(x) dx
3. Evaluate the integral. (Use C for the constant of
integration.)
cos−1(x) dx
4. Evaluate the integral. (Use C for the constant of
integration.)
ln(
x
) dx

Evaluate the integral using the indicated trigonometric
substitution. (Use C for the constant of integration.)
x3
x2 + 16
dx
, x = 4
tan(θ)

Question B:Consider the integral of sin(x) * cos(x) dx.
i) Do it using integration by parts; you might need the “break
out of the loop” trick. I would do u=sin(x), dv=cos(x)dx
ii) Do it using u-substitution. I would do u=cos(x)
iii) Do it using the identity sin(x)*cos(x)=0.5*sin(2x)
iv) Explain how your results in parts i,ii,iii relate to each
other.

Find this integral using Integration By Parts; use the Tabular
D.I. method.
∫ e^(5x) cos x dx

1. Find the antiderivative: indefinite integral(
sec^2(sqrt(x)dx
(a) State substitution. Use w for new variable
(b) Write new integral
(c) Write first step in solving new integral
(d) Write antiderivative answer
2. definite integral from 0 --> 1 (y + 1) / (e^(3y)) dy Leave
answer in exact form but simplify as much as possible
(a) Write problem in form so that integration by parts
applies.
(b) Write next step in solving integral
(c) Write answer in exact form

1. Al Einstein has a utility function that we can describe by
u(x1, x2) = x21 +
2x1x2 + x22
. Al’s wife, El Einstein, has a utility function v(x1,
x2) = x2 + x1.
(a) Calculate Al’s marginal rate of substitution between
x1 and x2.
(b) What is El’s marginal rate of substitution between
x1 and x2?
(c) Do Al’s and El’s utility functions u(x1,
x2) and v(x1, x2) represent the
same preferences?
(d) Is El’s utility function a...

Question about using the convolution of distribution:
1. we have the formula: integral fx(x)fy(z-x)dx=integral
fx(z-x)fy(x)dx
I know this are equivalent. However, how do I decide which side
I should use ?
For example,X~Exp(1) and Y~Unif [0,1] X and Y independnt and the
textbook use fx(z-x)fy(x)dx.
However, can I use the left hand side fx(x)fy(z-x)dx???is there
any constraint for using left or right or actually both can lead me
to the right answer???
2. For X and Y are independent and...

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