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Indefinite integrals

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Summary

Indefinite integrals

In a nutshell

Indefinite integrals are integrals that have a constant of integration. Indefinite integrals can be represented with an integration symbol.

The integration symbol

The integration symbol is $\int$ . This is used to integrate functions as follows:

$\int f(x) \, dx$

$\int$	The integration function.
$f(x)$	The function being integrated.
$dx$	The variable to integrate. In this case, integrate all functions of $x$ and ignore other letters.

Note: $dx$ can be read as "with respect to $x$ ".

Examples

Evaluate the following integrals:

i) $\int x^3\,dx$ 

ii) $\int (3\sqrt{x}+\dfrac{1}{x^2})\,dx$ 

iii) $\int (6y^2 +2ky+5)\,dy$

Part i):

Integrate $x^3$ using the rule:

$\underline{\int x^3 \, dx = \dfrac{x^4}{4}+C}$ 

Part ii):

Write each term in the form $ax^n$ and integrate them separately:

$\begin{aligned}\int (3\sqrt{x}+\dfrac{1}{x^2}) \, dx &= \int (3x^{\frac{1}{2}}+x^{-2})\,dx \\ &= \int 3x^{\frac12} \, dx + \int x^{-2} \, dx \\ &=\dfrac{3}{(\frac{3}{2})}x^{\frac{3}{2}}+\dfrac{1}{-1}x^{-1}+C\\ &= 2x^{\frac{3}{2}}-x^{-1}+C \end{aligned}$

$\underline{\int (3\sqrt{x}+\dfrac{1}{x^2}) \, dx=2x^{\frac{3}{2}}-x^{-1}+C}$

Part iii):

The $dy$ means to integrate with respect to $y$ .

$\begin{aligned}\int (6y^2 +2ky+5)\,dy &= \int 6y^2 \, dy + \int 2ky^1 \, dy + \int 5y^0\, dy \\&=\dfrac{6}{3}y^3+\dfrac{2k}{2}y^2+\dfrac{5}{1}y^1+C\\&=2y^3+ky^2+5y+C\end{aligned}$

$\underline{\int (6y^2 +2ky+5)\,dy=2y^3+ky^2+5y+C}$

Learn with Basics

Length:

Unit 1

Differentiating x^n

Unit 2

Integrating x^n

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Indefinite integrals

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Summary

Exercises

Select Lesson

Exam Board

Further kinematics

Application of forces

Projectiles

Forces and friction

Moments

Forces and motion

Variable acceleration

Constant acceleration

Modelling in mechanics

Normal distribution

Conditional probability

Correlation and hypothesis testing

Hypothesis testing I

Binomial distribution

Probability

Representation of data

Measures of location and spread

Data collection

Vectors II

Numerical methods

Integration II

Differentiation II

Trigonometry and modelling

Trigonometric functions

Radians

Sequences and series

Binomial expansion II

Parametric equations

Functions

Algebraic fractions

Proof II

Vectors I

Integration I

Differentiation I

Exponentials and logarithms

Trigonometric equations

Trigonometry

Binomial expansion I

Circles

Straight line graphs

Transformations

Sketching graphs

Polynomials

Equations and inequalities

Quadratics

Indices and surds

Proof I

Explainer Video

Summary

Indefinite integrals

In a nutshell

The integration symbol

Examples

Learn with Basics

Unit 1

Differentiating x^n

Unit 2

Integrating x^n

Jump Ahead

Unit 3