# 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: $$f(x) = x^n$$

Example: Evaluate the integral $\int x^5 dx$
Solution: $\int x^5 dx = \dfrac{x^{5 + 1}}{ 5 + 1} + c = \dfrac{x^6}{6} + c$

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

Example: Evaluate the integral $\int 5 \sin \; x dx$ Solution:
According to the above rule
$$\displaystyle \int 5 \sin (x) dx = 5 \int \sin(x) dx$$
$$\displaystyle \int \sin(x) dx$$   is given by 2.1 in table of integral formulas, hence
Hence
$$\displaystyle \int 5 \sin(x) dx = - 5 \cos x + C$$

## 3 - Integral of Sum of Functions.

Example: Evaluate the integral $\int (x + e^x) dx$
Solution:
According to the above property
$$\displaystyle \int (x + e^x) dx = \int x \; dx + \int e^x \; dx$$
$$\displaystyle \int x \; dx$$ is given by 1.3 and $$\displaystyle \int e^x \; dx$$ by 4.1 in table of integral formulas, hence
$\int (x + e^x) \; dx = \dfrac{x^2}{2}x + e^x + c$

## 4 - Integral of Difference of Functions.

Example: Evaluate the integral $\int (2 - 1/x) \; dx$
Solution:
According to the above property
$$\displaystyle \int (2 - 1/x) dx = \int 2 \; dx - \int (1/x) \; dx$$
$$\int 2 \; dx$$ is given by 1.2 and $$\int (1/x) \; dx$$ by 1.4 in table of integral formulas, hence
$\int (2 - 1/x) \; dx = 2x - \ln |x| + c$

## 5 - Integration by Substitution.

Example: Evaluate the integral $\int (x^2 - 1)^{20} 2x \; dx$
Solution:
Let $$u = x^2 - 1$$, hence $$du/dx = 2x$$ and the given integral can be written as
$$\displaystyle \int(x^2 - 1)^{20} \; 2x \; dx = \int u^{20} (du/dx) dx = \int u^{20} du$$
which evaluates to
$$= \dfrac{u^{21}}{21} + c$$
Substitute back
$$= \dfrac{(x^2 - 1)^{21}}{21} + c$$

## 6 - Integration by Parts.

Example: Evaluate the integral $\int \; x \; \cos x \; dx$
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,
$$\displaystyle \int \; x \cos x \; dx = x sin x - \int 1 \sin x dx$$
$$= x \sin x + \cos x + c$$

## 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. $$\displaystyle \int (1 / 2) \ln \; (x) dx$$
2. $$\displaystyle \int (\sin (x) + x^5 ) \; dx$$
3. $$\displaystyle \int (\sinh (x) - 3) \; dx$$
4. $$\displaystyle \int - x \sin (x) \; dx$$
5. $$\displaystyle \int \sin^{10}(x) \; \cos(x) dx$$

## 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:
$$\int$$(1 / 2) ln (x) dx = (1 / 2) $$\int$$ln (x) dx
We now use formula 4.3 in the table of integral formulas to evaluate $$\int$$ln (x) dx. Hence
$$\int$$(1 / 2) ln (x) dx = (1 / 2) ( (x ln (x)) - x ) + c

2.
Use rule 3 ( integral of a sum ) to obtain
$$\int$$[sin (x) + x 5] dx = $$\int$$ sin (x) dx + $$\int$$x 5 dx
We use formula 2.1 in the table of integral formulas to evaluate $$\int$$ sin (x) dx and rule 1 above to evaluate $$\int$$x 5 dx. Hence
$$\int$$[sin (x) + x 5] dx = - cos (x) + x 6 / 6
3.
Use rule 4 (integral of a difference) to obtain
$$\int$$(sinh (x) - 3) dx = $$\int$$ sinh (x) dx - $$\int$$3 dx
We use formula 7.1 in the table of integral formulas to evaluate $$\int$$ sinh (x) dx and integral of the constant 3 to obtain
$$\int$$(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
$$\int$$ - x sin (x) dx = - $$\int$$ f(x) g'(x) dx = - ( f(x) g(x) - $$\int$$ 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)) + $$\int$$ 1 (- cos(x)) dx
Use formula 2.2 in in the table of integral formulas to evaluate $$\int$$ 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
$$\int$$ sin10(x) cos dx = $$\int$$( u10 du/dx ) dx
Use rule 5 to write
= $$\int$$ u10 du
which gives
= u 11 / 11 + c
Substitute u by sin(x) to obtain
= (1 / 11) (sin 11(x) ) + c