[b]uGeneral rules[/u]/b

(\displaystyle\int{f'(x)} dx = f(x) + c )

(\displaystyle\int{\dfrac{dy}{dx}} dx = y + c )

When integrating sums and differences of functions, integrate them separately.

(\displaystyle\int{\big(f'(x) \pm g'(x)\big)} dx = \displaystyle\int{f'(x)} dx \pm \displaystyle\int{g'(x)} dx)

When integrating constant multiples of functions, integrate the function and multiply by the constant.

(\displaystyle\int{kf'(x)} dx = k\displaystyle\int{f'(x)} dx)

(\bm{\underline{x^n, \quad n \neq -1}} )

When integrating (xⁿ⁾ with respect to (x), increase the power (n) by (1), then divide by the new power (n + 1).

This works for all rational values of (n), i.e. fractions and negative values, except for (-1), because division by (0) is undefined.

Tip: Take care when increasing negative powers of (n) by (1), e.g. (-2+1=-1) not (-3)!

(\boxed{\displaystyle\int{x^n} dx = \dfrac{x^{n+1}}{n+1} + c, \quad n \neq -1} )

When integrating a number (k) with respect to (x), the result is (kx + c). Using the above result:

(\displaystyle\int{k} dx = \displaystyle\int{kx^0} dx = \dfrac{kx^1}{1} + c )

(\implies \boxed{\displaystyle\int{k} dx = kx + c} )

(\bm{\underline{x^n, n = -1}} )

(x^-1 = \dfrac{1}{x} )

[row]col(\boxed{\displaystyle\int{\dfrac{1}{x}} dx = \ln{x} + c} )/col
col(\boxed{\displaystyle\int{\dfrac{1}{kx}} dx = \dfrac{1}{k}\ln{x} + c} )[/col]/row
[b]uExponential functions[/u]/b

[row]col(\boxed{\displaystyle\int{e^x} dx = e^x + c} )/col
col(\boxed{\displaystyle\int{e^kx} dx = \dfrac{1}{k}e^{kx} + c} )[/col]/row
[b]uTrigonometric functions[/u]/b

[row]col(\boxed{\displaystyle\int{(\sin{x})} dx = -\cos{x} + c} )

(\boxed{\displaystyle\int{(\cos{x})} dx = \sin{x} + c} )

(\boxed{\displaystyle\int{(\sec^2{x})} dx = \tan{x} + c} )/col
col(\boxed{\displaystyle\int{(\sin{kx})} dx = -\dfrac{1}{k}\cos{kx} + c} )

(\boxed{\displaystyle\int{(\cos{kx})} dx = \dfrac{1}{k}\sin{kx} + c} )

(\boxed{\displaystyle\int{(\sec^2{kx})} dx = \dfrac{1}{k}\tan{kx} + c} )[/col]/row
Tip: Counterclockwise for integration.
mtaimg/images/topics/8/8-86-1.png/mtaimg

Trigonometric identities can be used to integrate more complex trigonometric functions.

For example, the topic=5/57power reduction formulae/topic can be used to integrate (\sin^2{x}) and (\cos^2{x}).

[b]uStandard integration results[/u]/b

(\displaystyle\int{x^n} dx = \dfrac{x^{n+1}}{n+1} + c, \quad n \neq -1 )

(\displaystyle\int{\dfrac{1}{kx}} dx = \dfrac{1}{k}\ln{x} + c )

(\displaystyle\int{e^kx} dx = \dfrac{1}{k}e^{kx} + c )

(\displaystyle\int{(\sin{kx})} dx = -\dfrac{1}{k}\cos{kx} + c )

(\displaystyle\int{(\cos{kx})} dx = \dfrac{1}{k}\sin{kx} + c )

(\displaystyle\int{(\sec^2{kx})} dx = \dfrac{1}{k}\tan{kx} + c )