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Indefinite Integration

Indefinite integration allows us to find antiderivatives of functions. It's essentially the reverse process of differentiation.

Basic Indefinite Integrals

Some common indefinite integrals that students should memorize are:

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$)
$\int \sin x dx = -\cos x + C$
$\int \cos x dx = \sin x + C$
$\int \frac{1}{x} dx = \ln |x| + C$
$\int e^x dx = e^x + C$

You can often derive them by using first principles.

Note

The '+C' at the end of each integral represents the constant of integration. This is crucial because indefinite integrals represent a family of functions that differ only by a constant.

Composites with Linear Functions

These basic integrals can be extended to include composites with linear functions of the form $ax + b$. For example:

$\int (ax + b)^n dx = \frac{(ax + b)^{n+1}}{a(n+1)} + C$ (where $n \neq -1$)
$\int \sin(ax + b) dx = -\frac{1}{a}\cos(ax + b) + C$
$\int \cos(ax + b) dx = \frac{1}{a}\sin(ax + b) + C$
$\int \frac{1}{ax + b} dx = \frac{1}{a}\ln |ax + b| + C$
$\int e^{ax + b} dx = \frac{1}{a}e^{ax + b} + C$

Example

Let's integrate $\int (2x + 3)^4 dx$:

Using the formula above with $a=2$, $b=3$, and $n=4$:

$\int (2x + 3)^4 dx = \frac{(2x + 3)^{4+1}}{2(4+1)} + C = \frac{(2x + 3)^5}{10} + C$

Reverse Chain Rule (Integration by Inspection)

The reverse chain rule is a technique used when we can spot a function and its derivative within the integrand. It's based on the chain rule of differentiation but applied in reverse.

The general form is:

$\int k g'(x) f(g(x)) dx = k F(g(x)) + C$

Where $F(x)$ is the antiderivative of $f(x)$ and $C$ is an arbitrary constant.

In other words, you look at it and find the antiderivative.

Example

Let's integrate $\int 2x(x^2+1)^4 dx$:

Here, we can spot $g(x) = x^2 + 1$ and $g'(x) = 2x$.

So, $k = 1$, $g'(x) = 2x$, and $f(g(x)) = (x^2+1)^4$.

The antiderivative of $x^4$ is $\frac{x^5}{5}$, so:

$\int 2x(x^2+1)^4 dx = \frac{(x^2+1)^5}{5} + C$

Integration by Substitution

Integration by substitution is a more formal method that can be used when the reverse chain rule is applicable. It involves making a substitution to simplify the integral.

The general steps are:

Choose a substitution $u = g(x)$
Find $\frac{du}{dx}$ and express $dx$ in terms of $du$
Rewrite the integral in terms of $u$
Integrate with respect to $u$
Substitute back to express the answer in terms of $x$

Example

Let's integrate $\int 4x \sin x^2 dx$:

Let $u = x^2$
Then $du = 2x dx$, or $dx = \frac{1}{2x}du$
Rewriting the integral: $\int 4x \sin x^2 \cdot \frac{1}{2x}du = \int 2 \sin u du$
Integrating: $-2 \cos u + C$
Substituting back: $-2 \cos x^2 + C$

Therefore, $\int 4x \sin x^2 dx = -2 \cos x^2 + C$

Common Mistake

Students often forget to substitute back to the original variable after integration. Always check if your final answer is in terms of the original variable!

Trigonometric Substitutions

Some integrals involving trigonometric functions can be solved using specific substitutions. One common example is:

$\int \frac{\sin x}{\cos x} dx$

Example

For the integral $\int \frac{\sin x}{\cos x} dx$, we use the substitution $u=\cos x$, $\frac{du}{dx} = -\sin x$

$$\int \frac{\sin x}{\cos x} dx = \int \frac{\sin x}{u} \frac{-1}{\sin x} du$$

$$\int \frac{\sin x}{\cos x} dx = -\int \frac{1}{u} du$$

$$\int \frac{\sin x}{\cos x} dx = -\ln (u) + C$$

$$\int \frac{\sin x}{\cos x} dx = -\ln (\cos x) + C$$

Tip

When dealing with trigonometric integrals, it's often helpful to recall trigonometric identities and relationships between different trigonometric functions.

Practice and Application

Mastering indefinite integration requires practice with a variety of problems. Students should work on recognizing patterns that suggest the use of the reverse chain rule or substitution.

Note

Remember, integration techniques are tools in your mathematical toolbox. The skill lies in recognizing which tool to use for each problem.

This diagram would illustrate how differentiation and integration are inverse operations, emphasizing the connection between the chain rule and the reverse chain rule.

Indefinite Integration

Indefinite integration allows us to find antiderivatives of functions. It's essentially the reverse process of differentiation.

Basic Indefinite Integrals

Some common indefinite integrals that students should memorize are:

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$)
$\int \sin x dx = -\cos x + C$
$\int \cos x dx = \sin x + C$
$\int \frac{1}{x} dx = \ln |x| + C$
$\int e^x dx = e^x + C$

You can often derive them by using first principles.

Note

The '+C' at the end of each integral represents the constant of integration. This is crucial because indefinite integrals represent a family of functions that differ only by a constant.

Composites with Linear Functions

These basic integrals can be extended to include composites with linear functions of the form $ax + b$. For example:

$\int (ax + b)^n dx = \frac{(ax + b)^{n+1}}{a(n+1)} + C$ (where $n \neq -1$)
$\int \sin(ax + b) dx = -\frac{1}{a}\cos(ax + b) + C$
$\int \cos(ax + b) dx = \frac{1}{a}\sin(ax + b) + C$
$\int \frac{1}{ax + b} dx = \frac{1}{a}\ln |ax + b| + C$
$\int e^{ax + b} dx = \frac{1}{a}e^{ax + b} + C$

Example

Let's integrate $\int (2x + 3)^4 dx$:

Using the formula above with $a=2$, $b=3$, and $n=4$:

$\int (2x + 3)^4 dx = \frac{(2x + 3)^{4+1}}{2(4+1)} + C = \frac{(2x + 3)^5}{10} + C$

Reverse Chain Rule (Integration by Inspection)

The reverse chain rule is a technique used when we can spot a function and its derivative within the integrand. It's based on the chain rule of differentiation but applied in reverse.

The general form is:

$\int k g'(x) f(g(x)) dx = k F(g(x)) + C$

Where $F(x)$ is the antiderivative of $f(x)$ and $C$ is an arbitrary constant.

In other words, you look at it and find the antiderivative.

Example

Let's integrate $\int 2x(x^2+1)^4 dx$:

Here, we can spot $g(x) = x^2 + 1$ and $g'(x) = 2x$.

So, $k = 1$, $g'(x) = 2x$, and $f(g(x)) = (x^2+1)^4$.

The antiderivative of $x^4$ is $\frac{x^5}{5}$, so:

$\int 2x(x^2+1)^4 dx = \frac{(x^2+1)^5}{5} + C$

Integration by Substitution

Integration by substitution is a more formal method that can be used when the reverse chain rule is applicable. It involves making a substitution to simplify the integral.

The general steps are:

Choose a substitution $u = g(x)$
Find $\frac{du}{dx}$ and express $dx$ in terms of $du$
Rewrite the integral in terms of $u$
Integrate with respect to $u$
Substitute back to express the answer in terms of $x$

Example

Let's integrate $\int 4x \sin x^2 dx$:

Let $u = x^2$
Then $du = 2x dx$, or $dx = \frac{1}{2x}du$
Rewriting the integral: $\int 4x \sin x^2 \cdot \frac{1}{2x}du = \int 2 \sin u du$
Integrating: $-2 \cos u + C$
Substituting back: $-2 \cos x^2 + C$

Therefore, $\int 4x \sin x^2 dx = -2 \cos x^2 + C$

Common Mistake

Students often forget to substitute back to the original variable after integration. Always check if your final answer is in terms of the original variable!

Trigonometric Substitutions

Some integrals involving trigonometric functions can be solved using specific substitutions. One common example is:

$\int \frac{\sin x}{\cos x} dx$

Example

For the integral $\int \frac{\sin x}{\cos x} dx$, we use the substitution $u=\cos x$, $\frac{du}{dx} = -\sin x$

$$\int \frac{\sin x}{\cos x} dx = \int \frac{\sin x}{u} \frac{-1}{\sin x} du$$

$$\int \frac{\sin x}{\cos x} dx = -\int \frac{1}{u} du$$

$$\int \frac{\sin x}{\cos x} dx = -\ln (u) + C$$

$$\int \frac{\sin x}{\cos x} dx = -\ln (\cos x) + C$$

Tip

When dealing with trigonometric integrals, it's often helpful to recall trigonometric identities and relationships between different trigonometric functions.

Practice and Application

Mastering indefinite integration requires practice with a variety of problems. Students should work on recognizing patterns that suggest the use of the reverse chain rule or substitution.

Note

Remember, integration techniques are tools in your mathematical toolbox. The skill lies in recognizing which tool to use for each problem.

This diagram would illustrate how differentiation and integration are inverse operations, emphasizing the connection between the chain rule and the reverse chain rule.

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Number and Algebra15 subtopics

Functions10 subtopics

Geometry and Trigonometry16 subtopics

Statistics and Probability19 subtopics

Calculus18 subtopics

AHL 5.11—Indefinite integration, reverse chain, by substitution Notes

Indefinite Integration

Basic Indefinite Integrals

Composites with Linear Functions

Reverse Chain Rule (Integration by Inspection)

Integration by Substitution

Trigonometric Substitutions

Practice and Application

Indefinite Integration

Number and Algebra15 subtopics

Functions10 subtopics

Geometry and Trigonometry16 subtopics

Statistics and Probability19 subtopics

Calculus18 subtopics

Indefinite Integration

Basic Indefinite Integrals

Composites with Linear Functions

Reverse Chain Rule (Integration by Inspection)

Integration by Substitution

Trigonometric Substitutions

Practice and Application

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Indefinite Integration

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