The problem was to find the indefinite integral of
______ Ö3x + 2 dx |
Step 1) Which of the rules that we developed in the last section can be applied to simplify this? Look at equation 11.2-11b. Can you see how this problem is just a special case of the general rule given in that equation? You just have to identify what the parts are. Equation 11.2-11b has the following parts that you have to identify with parts of the integral in this problem. They are: A, the constant by which the integrand is scaled; b, the constant by which x is scaled, and f, the constant by which bx is skewed. You also have to identify f, the function that is wrapped around all this.
Step 2) Identify A.
Since there is no scalar shown in this problem that multiplies over the entire integrand, that means that
Step 3) Identify b.
The thing in this problem that scales x is 3. So
Step 4) Identify f.
The thing that is added to bx (which in this case is 3x) is 2. So
Step 5) What is the function, f, that wraps this whole thing?
Clearly it's the square root function. The argument to that f is
_ f(u) = Öu
Step 6) What is an antiderivative of f? Look it up on the table if you don't already have that table memorized.
2 F(u) =u^{3/2} 3
Step 7) Put it all together using the rule from equation 11.2-11b.
That is put
______ 1 2 ^{ } 2 Ö3x + 2 dx = |
The problem was to find the indefinite integral of
(sin(px) + cos(3px)) dx |
Step 1: Recognize that this is a sum. That means that the rule in equation 11.2-13b applies. This rule breaks the integral into the sum of two simpler integrals:
(sin(px) + cos(3px)) dx = |
sin(px) dx + |
cos(3px) dx |
Step 2: Recognize that the rule in equation 11.2-11b applies to each of the summands.
So identify what the parameters of the rule are in each summand.
In both summands you have
Step 3: What are the f and g functions here that you can find in the table? Clearly if you let f be sin and g be cos, then what you have is
sin(px) + |
cos(3px) = |
f(px) + |
g(3px) |
Step 4: Apply the rule from equation 11.2-11b. In equations, that means that
f(px) + |
1 1 g(3px) = |
Step 5: Put in the functions for F and G. So you get as your answer
1 ^{ } 1 (sin(px) + cos(3px)) dx = - |
Step 1: Recognize that this is a sum. So again the rule in equation 11.2-13b applies. This rule breaks the integral into the sum of two simpler integrals:
2 ^{ } |
2 |
e^{-nx} dx |
Step 2: Recognize that the rule in equation 11.2-11b applies to each of the simpler integrals. So you need to determine A, b, and f for each of them. For the left integral, first turn it around into
2 |
e^{-nx} dx |
Step 3: What are the f and g functions here that you can find in the
table? The
F(x) = ln|x| G(x) = e^{x}
Step 4: Apply the rule in equation 11.2-11b to both summands. You have, according to the rule:
2f(-x+1) dx + |
1 g(-nx) dx = -2F(-x+1) - |
Step 5: Put in the functions for F and G.
2dx |
^{ } 1 e^{-nx} dx = -2ln|-x+1| - |
The problem was to find the indefinite integral,
x^{4} + 8x^{3} + 12x^{2} - 4x - 3 dx |
Step 1: Recognize that this (and every polynomial) is nothing more than the sum of power terms. Since the integral of a sum is the sum of the integrals, you can break this down into five separate integrals according to the rule in equation 11.2-13b. So the following is exactly equivalent to the original problem:
x^{4} dx + |
8x^{3} dx + |
12x^{2} dx - |
4x dx - |
3dx |
Step 2: Apply the rule in equation 11.2-3 to each integral that has a constant multiplier. This gives you
x^{4} dx + 8 |
x^{3} dx + 12 |
x^{2} dx - 4 |
x dx - 3 |
dx |
Step 3: Recognize that each of the integrals is now a case of x^{r}. And you can find that in the table. So apply that to each integral and you get
1 ^{ } 8 ^{ } 12 ^{ } 4 ^{ }I'll leave it to you to reduce the fractional coefficients to lowest terms on your own (if I were grading your paper, I wouldn't take off if you neglected to do that step -- after all, 12/3 is just as legitimate a way of expressing a coefficient as 4).x^{5} +x^{4} +x^{3} -x^{2} - 3x + C 5 _{ } 4 _{ } 3 _{ } 2 _{ }
The problem was to find the indefinite integral,
n å B_{k}(kx + 1)^{k} dx k=1 |
B_{k}(kx + 1)^{k}where the B_{k} is constant in every case. The hints told you that you needed only to find an antiderivative to this, and then you could put that back into the summation and have your answer. The rule in equation 11.2-11b tells you how to do this. The hints gave you a reason why, for the purposes of integration, you could treat k as a constant. That means that to set up the rule you have
1 F(x) =Taking the indefinite integral ofx^{k+1} k+1
B_{k} B_{k} f(kx + 1) dx = |
n B_{k} å^{ }If you didn't get this, go back over your work. If you still don't have a clue on this one but you got through the other exercises, then go back and review them, looking to see how what you applied in them is applied here.(kx + 1)^{k+1} + C k=1 k(k+1)
email me at hahn@netsrq.com