Question

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 (Don't add the constant of integration C since we have done it for you.) We have then found that ∫F(u)du= __________ +C and hence ∫cos3(x)esin(x)dx= __________ +C

Answer #1

Pat made the substitution x
−
3 = 2sin
t
in an integral and integrated to obtain
f(x)
dx
= 9t
−
2 sin
t
cos
t
+
C
.
Complete Pat's integration by doing the back substitution to find
the integral as a function of
x.
f(x) dx
=

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?

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

Identify the correct substitution (using u, du) to make in each
of the following integrals. Then, compute the integral.
a) ∫ 8x(x²-3)⁷dx
b) ∫ [√cot(x)] csc²(x) dx
c) ∫ cos(x)(sin³(x)+3sin²(x)a²+a³)dx where a is some
real number (Hint for C: Work smarter not harder)

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...

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

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.

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

Evaluate the integral: ∫27√x^2−9 / x^4 dx
(A) Which trig substitution is correct for this integral?
x=3tan(θ)
x=27sin(θ)
x=3sin(θ)
x=9tan(θ)
x=3sec(θ)
x=9sec(θ)
(B) Which integral do you obtain after substituting for xx and
simplifying?
Note: to enter θθ, type the word theta.
(C) What is the value of the above integral in terms of θ?
(D) What is the value of the original integral in terms of x?

Evaluate the integral: ∫8x^2 / √9−x^2 dx
(A) Which trig substitution is correct for this integral?
x=9sec(θ)
x=3tan(θ)
x=9tan(θ)
x=3sin(θ)
x=3sec(θ)
x=9sin(θ)
(B) Which integral do you obtain after substituting for x and
simplifying?
Note: to enter θθ, type the word theta.
(C) What is the value of the above integral in terms of θ?
(D) What is the value of the original integral in terms of x?

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