In Problems 45–50, the indefinite integral can be found in more than one way. First use the substitution method to find the indefinite integral. Then find it without using substitution. Check that your answers are equivalent . 49. ∫ 5 x 4 ( x 5 ) 4 d x
In Problems 45–50, the indefinite integral can be found in more than one way. First use the substitution method to find the indefinite integral. Then find it without using substitution. Check that your answers are equivalent . 49. ∫ 5 x 4 ( x 5 ) 4 d x
Solution Summary: The author explains how to calculate the value of the indefinite integral without using substitution method.
In Problems 45–50, the indefinite integral can be found in more than one way. First use the substitution method to find the indefinite integral. Then find it without using substitution. Check that your answers are equivalent.
49.
∫
5
x
4
(
x
5
)
4
d
x
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
The general solution of the linear system X' = AX is given.
A =
= (³ -2).
x(t) = c₁
c₁(1) et.
et + c₂
e-t
3
3
(a) In this case discuss the nature of the solution in a neighborhood of (0, 0).
O All solutions become unbounded and y = 3x serves as the asymptote.
O All solutions become unbounded and y = x serves as the asymptote.
If X(0) = X lies on the line y = x, then x(t) approaches (0, 0) along this line. Otherwise X(t) becomes unbounded and y = 3x serves as an
asymptote.
If X(0) = X lies on the line y = 3x, then x(t) approaches (0, 0) along this line. Otherwise x(t) becomes unbounded and y = x serves as an
asymptote.
O All solutions spiral toward (0, 0).
(b) With the aid of a calculator or a CAS, graph the solution that satisfies X(0) = (1, 1).
2
1
(1, 1)
x
-2
-1
1
2
4
-2
2
1
(1, 1)
4
2
-2
(1, 1)
2
x
4
-4
i
2
(1, 1)
1
x
1
2
2
1
1
2
x
Q2: Consider the problem
-((1+x)) = 0. x1 = [0, 1].
u(0) = 0, u'(1) = 1
Divide the interval / into three subintervals of equal length h -1/3 and let V), be the
corresponding space of continuous piecewise linear functions vanishing at x = 0.1.
Find the variational form and finite element method
Verify that the stiffness matrix A is given by:
16
1
-9
A = =
20
0
-11
11
Q2: A: Consider the problem
-Au+&u= f Χ Ε Ω, δ > 0
Χ Ε ΘΩ
Show that a(u, v) is continuous and V-elliptic.
B: Consider the model problem -u" f, xE 1 = 10, L. u(0) = u(L) = 0
Prove that u E Vo is solution of variational formulation if and only if its solution of
the minimization problem F(u) ≤ F(w) where F(w), w² dx
-
, fwdx
B-Solve the D.E of the following:
1- y+3y+2fy dt = f(t) for y(0)-1 if f(t) is the function
whose graph is shown below
2- y" +4y = u(t) for y(0)-y'(0)-0
3- y"+4y+13y=e-2t sin3t
1 2
for y(0)-1 and y'(0)=-2
Chapter 5 Solutions
Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.
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Definite Integral Calculus Examples, Integration - Basic Introduction, Practice Problems; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=rCWOdfQ3cwQ;License: Standard YouTube License, CC-BY