 Some functions may be changed to the standard forms by easy mathematical manipulation. To aid the process of changing to a recognisable form, use it also made of substitution, which results in a change of variable. In mathematics, an important use is made of what is called differentials. \begin{align} \displaystyle u &= f(x) \\ [...] # Trigonometric Integration by Substitution Substitution of Angle Parts Example 1 Find \displaystyle \int{(2x+3) \sin (x^2+3x)}dx. Show Solution \begin{align} \displaystyle \text{Let } u &= x^2+3x \\ \dfrac{du}{dx} &= 2x + 3 \\ du &= (2x+3)dx \\ \int{(2x+3) \sin (x^2+3x)}dx &= \int{\sin u}du \\ &= -\cos u + C \\ &= -\cos {(x^2+3x)} + C \\ \end{align} Substitution of [...] # Integration of Power Functions Home > Integration\displaystyle \int{(ax+b)^n}dx = \dfrac{(ax+b)^{n+1}}{a(n+1)}+cExample 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 [...] # Particular Values of Integration We can find the integral constant c, given a particular value of the function. Example 1 Find f(x) given that f'(x) = 3x^2+4x-5 and f(1)=3. Show Solution \begin{align} \displaystyle f(x) &= \int{(3x^2+4x-5)}dx \\ &= \dfrac{3x^{2+1}}{2+1} + \dfrac{4x^{1+1}}{1+1} - 5x +c \\ &= x^3 + 2x^2 - 5x + c \\ f(1) &= 3 \\ [...] # Integration using Double Angle Formula \begin{align} \displaystyle \cos{2x} &= 2\cos^2{x} - 1 \\ &= 1-2 \sin^2 {x} \\ &= \cos^2{x} - \sin^2{x} \\ \sin{2x} &= 2 \sin{x} \cos{x} \\ \end{align} These will be expressed in the following forms in order to apply to integrate. \begin{align} \displaystyle \sin^2 x &= \dfrac{1}{2}(1 - \cos 2x) \\ \cos^2 x &= [...] # 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 \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

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} [...] # Basic Integration Rules

Antiderivatives In many cases in calculus, it is known that the rate of change of one variable with respect to another, but we do not have a formula which relates the variables. In other words, it is known that $\dfrac{dy}{dx}$, but we need to know $y$ in terms of $x$. The process of finding $y$ [...] # Calculation of Areas under Curves

Consider the function $f(x)=x^2+2$. We wish to estimate the green area enclosed by $y=f(x)$, the $x$-axis, and the vertical lines $x=1$ and $x=4$. Suppose we divide the $x$-interval into three strips of width 1 unit. Upper Rectangles The diagram below shows upper rectangles, which are rectangles with top edges at the maximum value of the [...]