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Basics of integration for OCR A-level Maths

Basics of integration

This page covers the following topics:

1. Fundamental theorem of calculus
2. Integrating polynomials
3. Constant of integration
4. Indefinite integrals

The Fundamental Theorem of Calculus states that integration is the inverse process of differentiation. 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.

Fundamental theorem of calculus

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

Demonstrate the Fundamental Theorem of Calculus using the following equation: y = 10sinx + 6x + 10.

Differentiating the equation gives dy/dx = 10cosx + 6. Integrating this gives y = 10sinx + 6x + c, where c is the constant of integration. This shows that for c = 10, integrating has undone the differentiation, as stated by the Fundamental Theorem of Calculus.

Demonstrate the Fundamental Theorem of Calculus using the following equation: y = 10sinx + 6x + 10.

3

The integral of a function is sinx + 20x. Find the derivative of the function.

By the Fundamental Theorem of Calculus, differentiating the integral will give the function itself, therefore the function is y = cosx + 20. To find the derivative of the function, dy/dx = โˆ’sinx.

The integral of a function is sinx + 20x. Find the derivative of the function.

4

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.

5

Given that the integral of a function is 12xยณ + 21x + c, find the derivative of the function.

By the Fundamental Theorem of Calculus, differentiating the integral will give the function itself, therefore the function is y = 36xยฒ + 21. To find the derivative of the function, dy/dx = 72x.

Given that the integral of a function is 12xยณ + 21x + c, find the derivative of the function.

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