True or False? When integrating trigonometric functions, angles should be measured in degrees .

False. When integrating trigonometric functions, angles must be measured in radians .

What is reverse chain rule ?

Reverse chain rule , also known as integrating by inspection , is spotting that the chain rule could have been used to differentiate another function and turn it into the function you are trying to integrate.

What should be done with the integration limits when using substitution for a definite integral ?

When using substitution for a definite integral , the integration limits should be changed from x -values to u -values.

What is the final step in integration by substitution for an indefinite integral ?

The final step in integration by substitution for an indefinite integral is to substitute x (or whatever the original variable is) back in . The final answer should always be in terms of the original variable.

True or False? When using substitution for a definite integral , you must always substitute x back in before evaluating.

False. When using substitution for a definite integral, you can evaluate using the u limits without substituting x back in.

True or False? The constant of integration is needed in definite integration.

False. The constant of integration is not needed in definite integration.

What does negative integral refer to?

A negative integral is a definite integral that results in a negative value . This occurs when the area being calculated by the integral is below the x -axis.

True or False? The area under a curve is always positive.

True. The area under a curve is always positive , even if the definite integral is negative.

What should be done when finding the area under a curve that is partially below the x -axis ?

When finding the area under a curve that is partially below the x-axis , split the area into parts above and below the x -axis , calculate each integral separately, and sum the absolute values . If you are using a GDC to calculate the integral, then you can use the modulus version of the area under a curve integral. There is no need to split the area in this case.

True or False? When finding the area between a curve and a line , you always subtract the area under the line from the area under the curve .

False. When finding the area between a curve and a line , you may need to add or subtract areas depending on their relative positions.

True or False? When finding the area between a curve and a line , you always need to use integration to find any relevant areas.

False. When finding the area between a curve and a line , you don't always need to use integration to find relevant areas. For example, you may use basic area formulae (trapezoid, right triangle) to find the area under the line.

What should you do if no diagram is provided when solving area problems using integrals ?

If no diagram is provided when solving area problems using integrals , you should sketch a diagram, even if the curves are not completely accurate.

IBMathsDPAnalysis & Approaches (AA)SLFlashcards5. CalculusFurther Integration

Further Integration (DP IB Analysis & Approaches (AA))

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What is $\int \sin x d x$ ?

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What is $\int \sin x d x$ ?
$\int \sin x d x = - \cos x + C$
This is in the exam formula booklet.
What is $\int \cos x d x$ ?
$\int \cos x d x = \sin x + C$
This is in the exam formula booklet.
True or False?
$\int \sin (a x + b) = - \frac{1}{a} \cos (a x) + C$
False.
$\int \sin (a x + b) = - \frac{1}{a} \cos (a x + b) + C$
This is not in the exam formula booklet.
What is $\int e^{x} d x$ ?
$\int e^{x} d x = e^{x} + C$
This is in the exam formula booklet.
What is $\int \frac{1}{x} d x$ ?
$\int \frac{1}{x} d x = \ln |x| + C$
This is in the exam formula booklet.
True or False?
$\int e^{a x + b} d x = e^{a x + b} + C$ .
False.
$\int e^{a x + b} d x = \frac{1}{a} e^{a x + b} + C$ .
This is not in the exam formula booklet.
What is $\int \cos (a x + b) d x$ ?
$\int \cos (a x + b) d x = \frac{1}{a} \sin (a x + b) + C$
This is not in the exam formula booklet.
What is $\int \frac{1}{a x + b} d x$ ?
$\int \frac{1}{a x + b} d x = \frac{1}{a} \ln |a x + b| + C$
This is not in the exam formula booklet.
True or False?
$\int {(a x + b)}^{n} d x = \frac{1}{a (n + 1)} {(a x + b)}^{n + 1} + C$ , so long as $n \neq - 1$ .
True.
$\int {(a x + b)}^{n} d x = \frac{1}{a (n + 1)} {(a x + b)}^{n + 1} + C$ , so long as $n \neq - 1$ .
This is not in the exam formula booklet.
If $n \neq - 1$ , use $\int \frac{1}{a x + b} d x = \frac{1}{a} \ln |a x + b| + C$ .
True or False?
When integrating trigonometric functions, angles should be measured in degrees.
False.
When integrating trigonometric functions, angles must be measured in radians.
What is $\int \sec^{2} x d x$ ?
$\int \sec^{2} x d x = \tan x + C$
This is not in the exam formula booklet, but the standard derivative $f (x) = \tan x \Rightarrow f^{'} (x) = \sec^{2} x$ is in the formula booklet. The integral is just the inverse of that.
What is $\int \sec^{2} (a x + b) d x$ ?
$\int \sec^{2} (a x + b) d x = \frac{1}{a} \tan (a x + b) + C$
This is not in the exam formula booklet.
What is reverse chain rule?
Reverse chain rule, also known as integrating by inspection, is spotting that the chain rule could have been used to differentiate another function and turn it into the function you are trying to integrate.
True or false?
An integral in the form $\int \frac{f^{'} (x)}{f (x)} d x$ can be integrated with the reverse chain rule.
True.
An integral in the form $\int \frac{f^{'} (x)}{f (x)} d x$ can be integrated with the reverse chain rule, by using $\int \frac{f^{'} (x)}{f (x)} d x = \ln | f (x) | + C$ .
E.g. $\int \frac{2 x + 7}{x^{2} + 7 x - 12} d x = \ln |x^{2} + 7 x - 12| + C$
This is not in the exam formula booklet.
How can you integrate an integral of the form $\int g^{'} (x) f' (g (x)) d x$ with the reverse chain rule?
You can integrate an integral of the form $\int g^{'} (x) f' (g (x)) d x$ with the reverse chain rule by using $\int g^{'} (x) f' (g (x)) d x = f (g (x)) + C$ .
E.g. $\int 2 x \cos (x^{2}) d x = \sin (x^{2}) + C$ .
This is not in the exam formula booklet.
What is the "adjust and compensate" method for integration?
The adjust and compensate method is a technique used in reverse chain rule integration to deal with coefficients that don't match exactly.
E.g. rewriting $\int x \cos (x^{2}) d x$ as $\frac{1}{2} \int 2 x \cos (x^{2}) d x$ .
- the thing inside the integral has been adjusted by multiplying by 2
- $\frac{1}{2}$ is placed in front of the integral to compensate
$2 x \cos (x^{2})$ can be integrated directly using reverse chain rule.
What is integration by substitution?
Integration by substitution is a method used (e.g. when reverse chain rule is difficult to spot or awkward to use) to simplify an integral by rewriting it in terms of an alternative variable.
E.g. an integral in $x$ might be rewritten as a simpler integral in $u$ .
What is the first step in integration by substitution?
The first step in integration by substitution is to identify the substitution to be used.
When integrating a composite function, $f (g (x))$ , the substitution will frequently be $u$ equal to the secondary (or 'inside') function in the composite function, i.e $u = g (x)$ .
True or False?
In integration by substitution, $\frac{d u}{d x}$ can be treated like a fraction.
True.
In integration by substitution, $\frac{d u}{d x}$ can be treated like a fraction.
(Note: it is not actually a fraction, it is a gradient. But it works to treat it like a fraction in integrations like this.)
What would be a suitable substitution to use when integrating $\int (2 x + 5) e^{x^{2} + 5 x - 3} d x$ ?
When integrating $\int (2 x + 5) e^{x^{2} + 5 x - 3} d x$ , a suitable substitution would be $u = x^{2} + 5 x - 3$ .
Then $\frac{d u}{d x} = 2 x + 5 \Rightarrow d u = (2 x + 5) d x$ , so the integral can be rewritten as $\int e^{u} d u$ .
What should be done with the integration limits when using substitution for a definite integral?
When using substitution for a definite integral, the integration limits should be changed from x-values to u-values.
What is the final step in integration by substitution for an indefinite integral?
The final step in integration by substitution for an indefinite integral is to substitute x (or whatever the original variable is) back in.
The final answer should always be in terms of the original variable.
True or False?
When using substitution for a definite integral, you must always substitute x back in before evaluating.
False.
When using substitution for a definite integral, you can evaluate using the u limits without substituting x back in.
What is a definite integral?
A definite integral is an integral with specified upper and lower integration limits.
A definite integral is represented in the form $\int_{a}^{b} f (x) d x$
Where:
- $a$ is the lower integration limit
- $b$ is the upper integration limit
A definite integral can be evaluated using the equation $\int_{a}^{b} f (x) d x = F (b) - F (a)$ .
What is the function $F (x)$ , whose values appear on the right-hand side of the equation?
In the equation $\int_{a}^{b} f (x) d x = F (b) - F (a)$ , used to evaluate a definite integral, the function $F (x)$ is an antiderivative of $f (x)$ .
This is a result of the Fundamental Theorem of Calculus.
True or False?
The constant of integration is needed in definite integration.
False.
The constant of integration is not needed in definite integration.
True or false?
$\int_{a}^{b} k f (x) d x = k \int_{a}^{b} f (x) d x$
True.
$\int_{a}^{b} k f (x) d x = k \int_{a}^{b} f (x) d x$
This definite integrals property means that constant factors can be taken outside a definite integral.
What is $\int_{a}^{a} f (x) d x$ equal to?
$\int_{a}^{a} f (x) d x = 0$
The value of a definite integral with equal upper and lower limits is zero
True or False?
Swapping the limits of a definite integral doesn't change the result.
False.
Swapping the limits of a definite integral changes the sign of the result.
I.e. $\int_{b}^{a} f (x) d x = - \int_{a}^{b} f (x) d x$
Assuming $a \leq b \leq c$ , what is $\int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x$ equal to?
Assuming $a \leq b \leq c$ , then $\int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x = \int_{a}^{c} f (x) d x$ .
This definite integral property allows definite integrals to be 'split' into a sum of integrals over smaller intervals.
True or False?
$\int_{a}^{b} f (x) d x = \int_{a + k}^{b + k} f (x + k) d x$
False.
$\int_{a}^{b} f (x) d x = \int_{a - k}^{b - k} f (x + k) d x$ .
This definite integral property has to do with the horizontal translations of functions in definite integrals.
True or False?
$\int_{a}^{b} [f (x) + g (x)] d x = \int_{a}^{b} f (x) d x + \int_{a}^{b} g (x) d x$
True.
$\int_{a}^{b} [f (x) + g (x)] d x = \int_{a}^{b} f (x) d x + \int_{a}^{b} g (x) d x$
This definite integral property means you can rewrite the integral of a sum as a sum of integrals.
True or False?
$\int_{a}^{b} f (x) g (x) d x = (\int_{a}^{b} f (x) d x) (\int_{a}^{b} g (x) d x)$
False.
$\int_{a}^{b} f (x) g (x) d x$ is not equal to $(\int_{a}^{b} f (x) d x) (\int_{a}^{b} g (x) d x)$ .
You cannot rewrite the integral of a product as a product of integrals.
What does negative integral refer to?
A negative integral is a definite integral that results in a negative value.
This occurs when the area being calculated by the integral is below the x-axis.
True or False?
The area under a curve is always positive.
True.
The area under a curve is always positive, even if the definite integral is negative.
What is the formula for finding the area under a curve using the modulus function?
The formula for finding the area under a curve using the modulus function is $A = \int_{a}^{b} |y| d x$
Where:
- $A$ is the area being calculated
- $a$ and $b$ are the integration limits
- $y$ is the equation of the curve in terms of x
This formula is in the exam formula booklet.
What should be done when finding the area under a curve that is partially below the x-axis?
When finding the area under a curve that is partially below the x-axis,
- split the area into parts above and below the x-axis,
- calculate each integral separately,
- and sum the absolute values.
If you are using a GDC to calculate the integral, then you can use the modulus version of the area under a curve integral. There is no need to split the area in this case.
True or False?
When finding the area between a curve and a line, you always subtract the area under the line from the area under the curve.
False.
When finding the area between a curve and a line, you may need to add or subtract areas depending on their relative positions.
True or False?
When finding the area between a curve and a line, you always need to use integration to find any relevant areas.
False.
When finding the area between a curve and a line, you don't always need to use integration to find relevant areas.
For example, you may use basic area formulae (trapezoid, right triangle) to find the area under the line.
What is the general form of the integral used to find the area between two curves?
The general form of the integral used to find the area between two curves is $\int_{a}^{b} (y_{1} - y_{2}) d x$
Where
- $y_{1}$ is the upper function (the function 'on the top')
- $y_{2}$ is the lower function (the function 'on the bottom')
This formula is not given in the exam formula booklet.
True or False?
When finding the area between two curves, which curve is the 'upper' curve and which curve is the 'lower' curve never changes.
False.
When finding the area between two curves, which curve is the 'upper' curve and which is the 'lower' curve can change in different regions of the graph.
What should you do if no diagram is provided when solving area problems using integrals?
If no diagram is provided when solving area problems using integrals, you should sketch a diagram, even if the curves are not completely accurate.