IB Mathematics Applications & Interpretation Topic AHL 5.11—indefinite Integration, Reverse Chain, by Substitution Revision Notes (HL)

IB Mathematics Applications & Interpretation Topic AHL 5.11—indefinite Integration, Reverse Chain, by Substitution Revision Notes (HL) | RevisionDojo

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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 verify these results by differentiating the antiderivative.

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

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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 verify these results by differentiating the antiderivative.

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

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

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Number and Algebra15 subtopics

Functions10 subtopics

Geometry and Trigonometry16 subtopics

Statistics and Probability19 subtopics

Calculus18 subtopics

IB Mathematics Applications & Interpretation AHL 5.11—indefinite Integration, Reverse Chain, by Substitution Notes

Indefinite Integration

Basic Indefinite Integrals

Composites with Linear Functions

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Indefinite Integration

Basic Indefinite Integrals

Composites with Linear Functions

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Number and Algebra15 subtopics

Functions10 subtopics

Geometry and Trigonometry16 subtopics

Statistics and Probability19 subtopics

Calculus18 subtopics