Integration of Power Functions

Integration of Power Functions

Home > Integration $$\displaystyle \int{(ax+b)^n}dx = \dfrac{(ax+b)^{n+1}}{a(n+1)}+c$$ Example 1 Find $\displaystyle \int{(2x+1)^5}dx$. Show Solution \( \begin{align} \displaystyle \int{(2x+1)^5}dx &= \dfrac{(2x+1)^{5+1}}{2(5+1)} +c \\ &= \dfrac{(2x+1)^{6}}{12} +c \\ \end{align} \) Example 2 Find $\displaystyle \int{\dfrac{1}{(3x-2)^4}}dx$. Show Solution \( \begin{align} \displaystyle \int{\dfrac{1}{(3x-2)^4}}dx &= \int{(3x-2)^{-4}}dx \\ &= \dfrac{(3x-2)^{-4+1}}{3(-4+1)} +c\\ &= \dfrac{(3x-2)^{-3}}{-9} +c\\ &= -\dfrac{1}{9(3x-2)^3} +c\\ \end{align} \) Example 3 [...]
Integration of Rational Functions

Integration of Rational Functions

Integration of $\displaystyle \dfrac{1}{x}$ $$ \begin{align} \displaystyle \dfrac{d}{dx}\log_ex &= \dfrac{1}{x} \\ \log_ex &= \int{\dfrac{1}{x}}dx \\ \therefore \int{\dfrac{1}{x}}dx &= \log_ex +c\\ \end{align} $$ Example 1 Find $\displaystyle \int{\dfrac{2}{x}}dx$. Show Solution \( \begin{align} \displaystyle \int{\dfrac{2}{x}}dx &= 2\int{\dfrac{1}{x}}dx \\ &= 2\log_ex +c\\ \end{align} \) Example 2 Find $\displaystyle \int{\dfrac{1}{3x}}dx$. Show Solution \( \begin{align} \displaystyle \int{\dfrac{1}{3x}}dx &= \dfrac{1}{3}\int{\dfrac{1}{x}}dx \\ [...]
Integration of Trigonometric Functions

Integration of Trigonometric Functions

Integration of $\cos{x}$ $$ \begin{align} \displaystyle \dfrac{d}{dx}\sin{x} &= \cos{x} \\ \sin{x} &= \int{\cos{x}}dx \\ \therefore \int{\cos{x}}dx &= \sin{x} +c \\ \end{align} $$ Integration of $\cos{(ax+b)}$ $$ \begin{align} \displaystyle \dfrac{d}{dx}\sin{(ax+b)} &= \cos{(ax+b)} \times \dfrac{d}{dx}(ax+b) \\ &= \cos{(ax+b)} \times a \\ &= a\cos{(ax+b)} \\ \sin{(ax+b)} &= \int{a\cos{(ax+b)}}dx \\ &= a\int{\cos{(ax+b)}}dx \\ \dfrac{1}{a}\sin{(ax+b)} &= \int{\cos{(ax+b)}}dx \\ \therefore \int{\cos{(ax+b)}}dx [...]
Integration of Exponential Functions

Integration of Exponential Functions

Home > Integration The base formula of integrating exponential function is obtained from deriving $e^x$. $$ \begin{align} \displaystyle \dfrac{d}{dx}e^x &= e^x \\ e^x &= \int{e^x}dx \\ \therefore \int{e^x}dx &= e^x +c \\ \end{align} $$ This base formula is extended to the following general formula. $$ \begin{align} \displaystyle \dfrac{d}{dx}e^{ax+b} &= e^{ax+b} \times \dfrac{d}{dx}(ax+b) \\ &= e^{ax+b} [...]