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In what follows $c$ is a constant of integration, $f$, $u$ and $v$ are functions of $x$, $u '(x)$ and $v '(x)$ are the first derivatives of $u(x)$ and $v(x)$ respectively.
Basic Rules
- $\int a\,dx = a x + c$ (where $a$ is a constant)
- $\int a f(x)\,dx = a \int f(x)\,dx$ (where $a$ is a constant)
- Integration of constant power
$\int x^n \,dx = \dfrac{1}{n+1} x^{n+x} + c $ (for constant power $n$ not equal to $-1$)
- $\int {\dfrac{1}{x}} \,dx = \ln |x| + c $
- Integration of a sum
$\int [ u(x)+v(x)] \,dx = \int u(x)\,dx+\int v(x)\,dx$
- Integration of a difference
$\int [ u(x)-v(x)] \,dx = \int u(x)\,dx-\int v(x)\,dx$
- Integration using substitution
$\int f(u(x)) \cdot u'(x) \,dx = \int f(u)\,du$
Example 1: Evaluate the integral $\int x \cdot (x^2+5)^8 \,dx$
Let $u(x)=x^2+5$, hence $du/dx=2 x$ which gives $dx = du/2 x$
We now substitute to rewrite the given integral as
$\int x \cdot (x^2+5)^8 \,dx = \int \dfrac{1}{2 x} x \cdot u^8 \,du$
$= \int \dfrac{1}{2}\cdot u^8 \,du = (\dfrac{1}{2})\dfrac{1}{8+1} u^{8+1} + c = \dfrac{1}{18} (x^2+5)^9 + c$
- Integration by parts
$\int u(x)\cdot v'(x) \,dx = u(x) v(x) -\int u'(x) v(x) \,dx$
Example 1: Evaluate the integral $\int 2 x \cdot sin(x) \,dx$
Let $u(x)=2 x$ and $v'(x)=sin(x)$; hence $v(x) = -cos(x)$.
We now use the formula for integration by parts
$\int 2 x \cdot sin(x) \,dx = 2x (-cos(x)) - \int 2 (-cos(x)) \,dx=-2 x cos(x) + 2sin(x) + c$
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