Power Rule of Integration
$\int_{}^{} x^{n}dx = \frac{x^{n+1}}{n+1}$
(where c is the constant of integration)
Example 1: Integrate $x^{5}$ with respect to x.
Solution : $\int_{}^{} x^{5}dx$
By using the power rule
$\int_{}^{} x^{n}dx = \frac{x^{n+1}}{n+1}$
$\int_{}^{} x^{5}dx = \frac{x^{5+1}}{5+1}$
$\int_{}^{} x^{5}dx = \frac{x^{6}}{6} + c $ (where c is the constant of integration)
Example 2: Integrate $4x^{2}$ with respect to x.
Solution : $\int_{}^{} 4x^{2}dx$
By using the power rule
$\int_{}^{} x^{n}dx = \frac{x^{n+1}}{n+1}$
$\int_{}^{} 4x^{2}dx = \frac{4x^{2+1}}{2+1}$
$\int_{}^{} 4x^{2}dx = \frac{4x^{3}}{3} + c $ (where c is the constant of integration)
Example 3: Integrate $x^{-4}$ with respect to x.
Solution : $\int_{}^{} x^{5}dx$
By using the power rule
$\int_{}^{} x^{n}dx = \frac{x^{n+1}}{n+1}$
$\int_{}^{} x^{-4}dx = \frac{x^{-4+1}}{-4+1}$
$\int_{}^{} x^{-4}dx = \frac{x^{-3}}{-3}$
$\int_{}^{} x^{-4}dx = \frac{-1}{3x^{3}}$ + c (where c is the constant of integration)
Example 4: Integrate $\sqrt{x}$ with respect to x.
Solution : $\int_{}^{} \sqrt{x}dx$
By using the power rule
$\int_{}^{} x^{n}dx = \frac{x^{n+1}}{n+1}$
$\int_{}^{} \sqrt{x}dx =\int_{}^{} x^\frac{1}{2}dx$
$\int_{}^{} x^\frac{1}{2}dx =\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}$
$\int_{}^{} x^\frac{1}{2}dx=\frac{x^\frac{3}{2}}{\frac{3}{2}}$
$\int_{}^{} x^\frac{1}{2}dx=\frac{2x^\frac{3}{2}}{3}$
$\int_{}^{} x^\frac{1}{2}dx=\frac{2\sqrt{x^{3}}}{3}$ + c