∫(f + g) dx = ∫f dx + ∫g dx. (𝑥 ) 𝑥 =𝐹( )−𝐹( )=lim𝑥→ −𝐹𝑥− lim𝑥→ +𝐹(𝑥) )odd function: If < < , and ( )is undefined, then ∫ (𝑥) 𝑥 =
2.1.11.12.4 Chapter 4 Indefinite Integrals
If n6= 1 lnjxj+ c;
B() () () () a.
Ap calculus integration rules 1. Power rule (n≠−1) ∫ x n dx: Apr 11 6:04 pm (3 of 15) title: Apr 11 6:06 pm (6 of 15) title:
U n du u e du 4.
∫x n dx = (x n+1 /n+1) + c. A constant rule, a power rule, linearity, and a limited few rules for trigonometric, logarithmic, and exponential functions. ( 6 9 4 3)x x x dx32 3 3. Answer will be negative (+5) !×!
If n= 1 exponential functions with base a:
If (𝑥=− (−𝑥), then ∫ (𝑥) 𝑥 − =0 undefined points: General integration deflnitions and methods: As per the power rule of integration, if we integrate x raised to the power n, then; Then ( ) (*) 1 lim i b n a n i f x dx f x x →∞ = ∫ =∑ ∆.
∫ (f + g) dx:
© 2005 paul dawkins chain rule variants the chain rule applied to. ∫ f dx + ∫ g dx: When m+n is a negative even integer then put tan x = t. Integrals with trigonometric functions z sinaxdx= 1 a cosax (63) z sin2 axdx= x 2 sin2ax 4a (64) z sinn axdx= 1 a cosax 2f 1 1 2;
The function is a product of two or more functions, at least one of which can be integrated.
Apr 11 6:05 pm (4 of 15) title: Dx x xx 1 5. Integral calculus formula sheet derivative rules: Apr 11 6:07 pm (7 of 15).
If m is an odd natural number then put cos x = t.
Òòcf(x)dx= cf(x)dx, cis a constant. ( ) 3 x dx 3 2;cos2 ax (65) z. Use if the function to be integrated is a basic function, or if it can be rewritten as a basic function.
These are some of the most frequently encountered rules for differentiation and integration.
Apr 11 6:05 pm (5 of 15) title: Suppose f x( ) is continuous on [ab,]. By this rule the above integration of squared term is justified, i.e.∫x 2 dx. Theorem let f(x) be a continuous function on the interval [a,b].
1 ( ) ( ) = ( ) 1 ( ) 1 ( ^ ( ) 1 ( ) ) to decide first function.
We can use this rule,. ( 2 3)x x dx 2 23 8 5 6 4. (5 8 5)x x dx2 2. Where kis a constant z xn dx= 1 n+ 1 xn+1 + c;
The function to be integrated becomes simpler when we replace some portion of the function with u.
If n6= 1 z 1 x dx= lnjxj+ c z kf(x)dx= k z f(x)dx z (f(x) g(x))dx= Divide [ab,] into n subintervals of width ∆x and choose * x i from each interval. The important rules for integration are: Besides that, a few rules can be identi ed:
Fundamental rules ( ) 𝑥 =0 ∫ 𝑥=𝑥+𝐶
Common integrals indefinite integral method of substitution ∫ ∫f g x g x dx f u du( ( )) ( ) ( )′ = integration by parts ∫ ∫f x g x dx f x g x g x f x dx( ) ( ) ( ) ( ) ( ) ( )′ ′= − integrals of rational and irrational functions 1 1 n x dx cn x n + = + ∫ + 1 dx x cln x ∫ = + ∫cdx cx c= + 2 2 x ∫xdx c= + 3 2 3 x ∫x dx c= + Now use the rules for adding: Xn+1 n+ 1 + c; © 2005 paul dawkins integrals definitions definite integral:
X n+1 n+1 + c:
If both m and n are odd natural numbers then put either sin x = t or cos x = t. Z ax dx= ax ln(a) + c with base e, this becomes: For the following, let u and v be functions of x, let n be an integer, and let a, c, and c be constants. If n is an odd natural number then put sin x = t.
0 d c dx nn 1 d xnx dx sin cos d x x dx sec sec tan d x xx dx tan sec2 d x x dx cos sin d x x dx csc csc cot d x xx dx cot csc2 d x x dx d aaaxxln dx d eex x dx dd cf x c f x dx dx
We use i inverse (example ^( 1) ) l log (example log ) a algebra (example x2, x3) t trignometry (example sin2 x) e exponential (example ex) 2. Basic integration formulas and the substitution rule 1the second fundamental theorem of integral calculus recall fromthe last lecture the second fundamental theorem ofintegral calculus. Z ex dx= ex + c if we have base eand a linear function in the exponent, then z eax+b dx= 1 a eax+b + c trigonometric functions z Where stands for nth differential coefficient of u and stands for nth integral of v.
Integral of a constant \int f\left(a\right)dx=x\cdot f\left(a\right) take the constant out \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx.
Integration rules and techniques antiderivatives of basic functions power rule (complete) z xn dx= 8 >> < >>:
