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7.1 Integration by Substitution (Part I)

1. Knowing the chain rule for differentiation is helpful for finding antiderivatives.
2. An incorrect guess cannot help find the correct antiderivative.
3. When making a substitution in the integral of 3x^2 cos(x^3), the "inside function" is the cosine.
4. The method of substitution is a formalization of the guess-and-check method.
5. The method of substitution always works when there is an inside function and outside function.
6. A common pattern for an integral is to have a function in the denominator and its derivative in the numerator.

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7.2 Integration by Substitution (Part II)

1. After making a substitution, the original limits of integration are still used.
2. In Example 2, the substitution made is w = cos theta .
3. Sometimes you must solve the substitution w = f(x) for x before finding dx.
4. The integral of sqrt(1-x^2) from -1 to 1 is evaluated using the Fundamental Theorem of Calculus in this section.
5. The area of the ellipse x^2/a +y^2/b = 1 is pi/ab.

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7.3 Integration by Parts

1. The first guess of x*e^x as an antiderivative of x*e^x can be corrected by multiplying by a constant.
2. Integration by parts comes from the chain rule for derivatives.
3. Integration by parts always evaluates an integral directly.
4. It helps if u' is simpler than u and v is simpler than v'.
5. Integration by parts does not apply to definite integrals.
6. It may be necessary to apply integration by parts more than once.

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7.4 Tables of Integrals

1. Most functions have elementary antiderivatives.
2. One section in the table of integrals is "Cubes in the Denominator."
3. "Reduction formulas" may be used repeatedly.
4. The polynomial x^2+6x+14 is a perfect square.
5. Partial fractions requires the decomposition of a quotient intotwo or more fractions and comes from the solution of several equations inseveral unknowns.

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7.5 Approximating Definite Integrals

1. Riemann sums are only calculated using left sums and right sums.
2. The midpoint rule is the average of the left and right sums.
3. The trapezoid rule gives an over estimate if the integral is concave up.
4. The midpoint rule gives an overestimate if the integral is concave up.
5. The trapezoid in Figure 7.6 has top edge drawn as a secant line.

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7.6 Approximation Errors and Simpson's Rule

1. Using left or right sums, each extra digit of accuracy requires 10 times as many subintervals.
2. Using the trapezoid or midpoint rules, each extra digit of accuracy requires 10 times as many subintervals.
3. If it takes a calculator half a second to evaluate a midpoint approximately accurate to 4 digits, then it will take 50 seconds to increase that approximation to 6 digits.
4. The error in the trapezoid rule is half that of the midpoint rule.
5. In Simpson's rule, each extra 4 digits of accuracy requires about 10 times as much work.
6. Errors using Simpson's rule are proportional to n^4.

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7.7 Improper Integrals

1. Improper integrals have an infinite limit or an unbounded integrand.
2. Taking a hydrogen atom apart requires moving the electron to an infinite distance from the proton.
3. It is possible to have an improper integral with both limits being (positive and negative) infinity.
4. An integrand can only become infinite at a one of the limits of integration.
5. An integral that doesn't converge is said to polyverge.

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7.8 More on Improper Integrals

1. You may determine that an improper integral converges without knowing its value.
2. To show convergence, f(x), the integrand, needs to be larger than one whose integral converges.
3. The integral of 1/x^p from 1 to infinity converges for p >1 and diverges for p < 1.
4. The integral of (sin x + 3)/sqrt(x) from 1 to infinity diverges since the integral of 2/sqrt(x) from 1 to infinity diverges and (sin x + 3)/sqrt(x) > 2/sqrt(x).
5. In comparing two integrands, one must be larger than the other over the entire interval of integration to decide about convergence.