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Basics of integration for SQA Higher Maths

Basics of integration

This page covers the following topics:

1. Integrating polynomials
2. Constant of integration
3. Indefinite integrals

The integral of a polynomial ax^n is given by ax^(n + 1)/(n + 1) + c, where c is the constant of integration.

Integrating polynomials

During differentiation, the constant terms of the function being differentiated will disappear. Thus, to account for this in integration, a constant is added to the integral. This constant is usually expressed as c and is called the constant of integration.

Constant of integration

Indefinite integrals do not have an evaluated constant of integration. They represent a family of functions which have the same derivative, and their only difference is the value of the constant of integration.

Indefinite integrals

1

A gradient function is given in the graph. Find the general form of the equations whose derivative follows the given gradient function.

These equations are the indefinite integrals of the gradient function. So, integrating gives y = xโด/4 + 5xยณ/3 โˆ’ 5xยฒ + 4x + c, where c is the constant of integration. This is the equation form of all functions whose derivative follows the gradient function.

A gradient function is given in the graph. Find the general form of the equations whose derivative follows the given gradient function.

2

Evaluate โˆซ(10x + 8)dx.

The integral is given by 5xยฒ + 8x + c, where c is the constant of integration.

5xยฒ + 8x + c

Evaluate โˆซ(10x + 8)dx.

3

Solve the integral โˆซ(4x + 32xโท)dx.

The integral is given by 2xยฒ + 4xโธ + c, where c is the constant of integration.

2xยฒ + 4xโธ + c

Solve the integral โˆซ(4x + 32xโท)dx.

4

Define indefinite integrals.

Indefinite integrals are integrals whose constant of integration is not evaluated. They represent a family of functions which have the same derivative, and their only difference is the value of the constant of integration.

Define indefinite integrals.

5

Evaluate โˆซ(10x + 8)dx.

The integral is given by 5xยฒ + 8x + c, where c is the constant of integration.

5xยฒ + 8x + c

Evaluate โˆซ(10x + 8)dx.

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