Power Rule of Integration

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The power rule of integration gives us the formula that allows us to integrate any function that can be written as a power (exponent) of x. In general we can write this power function as $x^{n}$.
Sometimes the exponent of x may be negative, fractional or sometimes the integrand (the term to integrate) may be in radical.
The power rule of integration is an necessary step in integration or it is the basic step of integration.

Power Rule of Integration
$\int_{}^{} x^{n}dx = \frac{x^{n+1}}{n+1}$
(where c is the constant of integration)

Examples on Power Rule 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

12th grade math

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