# Rules of Integrals with Examples

A tutorial, with examples and detailed solutions, in using the rules of indefinite integrals in calculus is presented. A set of questions with solutions is also included.

In what follows, C is a constant of integration and can take any value.

## 1 - Integral of a power function: f(x) = x ^{n}

^{n}dx = x

^{ n + 1 }/ (n + 1) + c

__Example:__Evaluate the integral

^{5}dx

__Solution:__

^{5}dx = x

^{5 + 1}/ ( 5 + 1) + c = x

^{6}/ 6 + c

## 2 - Integral of a function f multiplied by a constant k: k f(x)

__Example:__Evaluate the integral

__Solution:__

According to the above rule

∫ 5 sin (x) dx = 5 ∫ sin(x) dx

∫ sin(x) dx is given by 2.1 in table of integral formulas , hence

∫ 5 sin(x) dx = - 5 cos x + C

## 3 - Integral of Sum of Functions.

__Example:__Evaluate the integral

^{ x}] dx

__Solution:__

According to the above property

∫ [x + e

^{ x}] dx = ∫ x dx + ∫ e

^{ x}dx

ò x dx is given by 1.3 and ∫ e

^{ x}dx by 4.1 in table of integral formulas , hence

∫ [x + e

^{ x}] dx = x

^{ 2}/ 2 + e

^{ x}+ c

## 4 - Integral of Difference of Functions.

__Example:__Evaluate the integral

__Solution:__

According to the above property

∫ [2 - 1/x] dx = ∫ 2 dx - ∫ (1/x) dx

∫ 2 dx is given by 1.2 and ∫ (1/x) dx by 1.4 in table of integral formulas , hence

∫ [2 - 1/x] dx = 2x - ln |x| + c

## 5 - Integration by Substitution.

__Example:__Evaluate the integral

^{ 2}- 1)

^{ 20}2x dx

__Solution:__

Let u = x

^{ 2}- 1, du/dx = 2x and the given integral can be written as

∫ (x

^{ 2}- 1)

^{ 20}2x dx

= ∫ u

^{ 20}(du/dx) dx

= ∫ u

^{ 20}du according to above property

= u

^{ 21}/ 21 + c = (x

^{ 2}- 1)

^{ 21}/ 21 + c

## 6 - Integration by Parts.

__Example:__Evaluate the integral

__Solution:__

Let f(x) = x and g ' (x) = cos x which gives

f ' (x) = 1 and g(x) = sin x

From integration by parts formula above,

∫ x cos x dx = x sin x - ∫ 1 sin x dx

= x sin x + cos x + c

## More Questions with Solutions

Use the table of integral formulas and the rules above to evaluate the following integrals. [Note that you may need to use more than one of the above rules for one integral].1. ∫ (1 / 2) ln (x) dx

2. ∫ [sin (x) + x

^{ 5}] dx

3. ∫ [sinh (x) - 3] dx

4. ∫ - x sin (x) dx

5. ∫ sin

^{10}(x) cos(x) dx

## Solutions to the Above Questions

1.

This is the integral of ln (x) multiplied by 1 / 2 and we therefore use rule 2 above to obtain:

∫ (1 / 2) ln (x) dx = (1 / 2) ∫ ln (x) dx

We now use formula 4.3 in the table of integral formulas to evaluate ∫ ln (x) dx. Hence

∫ (1 / 2) ln (x) dx = (1 / 2) ( (x ln (x)) - x ) + c

2.

Use rule 3 ( integral of a sum ) to obtain

∫ [sin (x) + x

^{ 5}] dx = ∫ sin (x) dx + ∫ x

^{ 5}dx

We use formula 2.1 in the table of integral formulas to evaluate ∫ sin (x) dx and rule 1 above to evaluate ∫ x

^{ 5}dx. Hence

∫ [sin (x) + x

^{ 5}] dx = - cos (x) + x

^{ 6}/ 6

3.

Use rule 4 (integral of a difference) to obtain

∫ (sinh (x) - 3) dx = ∫ sinh (x) dx - ∫ 3 dx

We use formula 7.1 in the table of integral formulas to evaluate ∫ sinh (x) dx and integral of the constant 3 to obtain

∫ (sinh (x) - 3) dx = cosh (x) - 3 x + c

4.

The integrand is the product of two function x and sin (x) and we try to use integration by parts in rule 6 as follows:

Let f(x) = x , g'(x) = sin(x) and therefore g(x) = - cos(x)

Hence

∫ - x sin (x) dx = - ∫ f(x) g'(x) dx = - ( f(x) g(x) - ∫ f'(x) g(x) dx)

Substitute f(x), f'(x), g(x) and g'(x) by x , 1, sin(x) and - cos(x) respectively to write the integral as

= - x (- cos(x)) + ∫ 1 (- cos(x)) dx

Use formula 2.2 in in the table of integral formulas to evaluate ∫ cos(x) dx and simplify to obtain

= x cos (x) - sin(x) + c

5.

Let u = sin(x) and therefore du/dx = cos(x). Hence the given integral can be written as

∫ sin

^{10}(x) cos dx = ∫ ( u

^{10}du/dx ) dx

Use rule 5 to write

= ∫ u

^{10}du

which gives

= u

^{11}/ 11 + c

Substitute u by sin(x) to obtain

= (1 / 11) (sin

^{11}(x) ) + c

## More References and Links

Table of Integral Formulasintegrals and their applications in calculus.

evaluate integrals .

Integration by Substitution .