Integrating simple algebraic expressions

Integration is the inverse process to differentiation. Some people call it anti-differentiation.

Instead of multiplying the power at the front and subtracting one from the power, we add one to the power and then divide by the new power.

Example

\int {{x^2}}\,\, dx

Solution

This just means, integrate {x^2} with respect to x. Remember, add one to the power and divide by the new power.

\int {{x^2}}\,\, dx

= \frac{{{x^3}}}{3} + c

The + c appears because when you differentiate a constant term, the answer is zero, so as we are performing 'anti-differentiation', we presume there may have been a constant term, which reduced to zero when differentiated. This c is called the constant of integration.

In general:

\frac{{dy}}{{dx}} = a{x^n} \to y = \frac{{a{x^{n + 1}}}}{{n + 1}} + c provided n \ne  - 1

curriculum-key-fact
  • In other words you add one to the power, divide by the new power and add the constant of integration.

Question

Find \int {({x^4}}  + {x^3})\,\,dx

\int {({x^4}}  + {x^3})\,\,dx

= \frac{{{x^5}}}{5} + \frac{{{x^4}}}{4} + c

Question

Find \int {(4{x^3}}  + 7{x^{ - 2}})\,\,dx

\int {(4{x^3}}  + 7{x^{ - 2}})\,\,dx

= \frac{{4{x^4}}}{4} + \frac{{7{x^{ - 1}}}}{{ - 1}} + c

= {x^4} - \frac{7}{x} + c

Question

Find \int {{{(x + 2)}^2}}\,\,dx

Similar rules apply to integration whereby we need to remove the brackets first as the expression has to be sums and/or differences of terms of the form a{x^n}.

\int {{{(x + 2)}^2}}\,\, dx

= \int {({x^2}}+ 4x + 4)\,\,dx

= \frac{{{x^3}}}{3} + \frac{{4{x^2}}}{2} + 4x + c

= \frac{{{x^3}}}{3} + 2{x^2} + 4x + c

Question

Find \int {\frac{{x + \sqrt x  + \sqrt[3]{x}}}{x}}\,\,dx

\int {\frac{{x + \sqrt x  + \sqrt[3]{x}}}{x}}\,\,dx

= \int {\frac{{x + {x^{\frac{1}{2}}} + {x^{\frac{1}{3}}}}}{x}}\,\,dx

= \int {\frac{x}{x}}  + \frac{{{x^{\frac{1}{2}}}}}{x} + \frac{{{x^{\frac{1}{3}}}}}{x}\,\,dx

= \int {(1 + {x^{ - \frac{1}{2}}}}  + {x^{ - \frac{2}{3}}})\,\,dx

= x + \frac{{{x^{\frac{1}{2}}}}}{{\frac{1}{2}}} + \frac{{{x^{\frac{1}{3}}}}}{{\frac{1}{3}}} + c

= x + 2\sqrt x  + 3\sqrt[3]{x} + c