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Integration is the inverse of differentiation.

It is the process of finding the function whose derivative is known.

Because integration is the inverse of differentiation, it is also known as anti-derivative.

One way of integrating is using the power rule.

The power of the rule of integration is easily achieved by adding one to the exponent and then dividing by the value of the added exponent.

That is,

$$\int x dx=\frac{x^{n+1}}{n+1}$$

The power rule of integration allows us to integrate any function that can be written as a power of x.

And, always remember to add C. This ensures that the function derive can account for all possible solutions of the integral.

Example 1

Evaluate $\int 8x^5 dx$ 


By power rule,



Example 2

Evaluate $\int 4x^2+3x dx$


By power rule



Example 3

Integrate $4x^3+6x^2+2$


$\int 4x^3+6x^2+2dx$

By power rule




Example 4

Integrate $(1-x)^2$


Before integrating, let's expand the expression



We can now integrate

$\int 1-2x+x^2 dx$




Example 5

Evaluate $\int \left(\frac{2t^3-3t}{4t}\right) dx $


Before integrating, let's simply the expression further



Let's now integrate

$ \int \left ({t^3}{4}-\frac{3}{4}\right) + C $

By power rule




Example 6

Evaluate $\int \left(\frac{(3x^2-2)^2}{x^2}\right) dx$


Before we integrate, let first simplify.


Expanding the numerator




Splitting the fraction


By law of indices,



Having simplify, we can now integral

$\int 9x^2-12+4x^{-2} dx$

Using the power rule



If you like, you can express in fractional form


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