Thefartiste – Blog reviews everything Thefartiste.com is a website that reviews and synthesizes knowledge and the most attractive and cheap quality products to our readers.
Integration Rules
- Integration
- Examples
- Example: what is the integral of sin(x) ?
- Example: what is the integral of 1/x ?
- Power Rule
- Example: What is ∫x3 dx ?
- Example: What is ∫√x dx ?
- Multiplication by constant
- Example: What is ∫6x2 dx ?
- Sum Rule
- Example: What is ∫(cos x + x) dx ?
- Difference Rule
- Example: What is ∫(ew − 3) dw ?
- Sum, Difference, Constant Multiplication And Power Rules
- Example: What is ∫(8z + 4z3 − 6z2) dz ?
- Integration by Parts
- Substitution Rule
- Final Advice
Integration
consolidation can be used to find areas, volumes, central points and many utilitarian things. It is much used to find the area underneath the graph of a function and the x-axis .
The first dominion to know is that integrals and derivatives are opposites !
sometimes we can work out an built-in,
because we know a duplicate derived function.
Reading: Integration Rules
Integration Rules
here are the most utilitarian rules, with examples below :
| Common Functions | Function | Integral |
|---|---|---|
| Constant | ∫a dx | ax + C |
| Variable | ∫x dx | x2/2 + C |
| Square | ∫x2 dx | x3/3 + C |
| Reciprocal | ∫(1/x) dx | ln|x| + C |
| Exponential | ∫ex dx | ex + C |
| ∫ax dx | ax/ln(a) + C | |
| ∫ln(x) dx | x ln(x) − x + C | |
| Trigonometry (x in radians) | ∫cos(x) dx | sin(x) + C |
| ∫sin(x) dx | -cos(x) + C | |
| ∫sec2(x) dx | tan(x) + C | |
| Rules | Function | Integral |
| Multiplication by constant | ∫cf(x) dx | c ∫f(x) dx |
| Power Rule (n≠−1) | ∫xn dx | xn+1 n+1 + C |
| Sum Rule | ∫(f + g) dx | ∫f dx + ∫g dx |
| Difference Rule | ∫(f – g) dx | ∫f dx – ∫g dx |
| Integration by Parts | See Integration by Parts | |
| Substitution Rule | See Integration by Substitution | |
Examples
Example: what is the integral of sin(x) ?
From the board above it is listed as being −cos(x) + C
It is written as :
∫sin ( x ) dx = −cos ( x ) + C
Example: what is the integral of 1/x ?
From the table above it is listed as being ln|x| + C
It is written as :
∫ ( 1/x ) dx = ln|x| + C
The erect bars || either side of x mean absolute respect, because we do n’t want to give negative values to the natural logarithm function ln .
Power Rule
Example: What is ∫x3 dx ?
The interview is asking “ what is the integral of x3 ? ”
We can use the Power Rule, where n=3 :
∫xn dx = xn+1 n+1 + C
∫x3 dx = x4 4 + C
Example: What is ∫√x dx ?
√x is besides x0.5
We can use the Power Rule, where n=0.5 :
∫xn dx = xn+1 n+1 + C
∫x0.5 dx = x1.5 1.5 + C
Multiplication by constant
Example: What is ∫6×2 dx ?
We can move the 6 outside the integral :
∫6×2 dx = 6∫x2 dx
And immediately use the Power Rule on x2 :
= 6 x3 3 + C
simplify :
= 2×3 + C
Sum Rule
Example: What is ∫(cos x + x) dx ?
Use the Sum convention :
∫ ( cobalt x + x ) dx = ∫cos x dx + ∫x dx
study out the integral of each ( using mesa above ) :
= sine x + x2/2 + C
Difference Rule
Example: What is ∫(ew − 3) dw ?
Use the Difference rule :
∫ ( electronic warfare − 3 ) dw =∫ew dw − ∫3 dw
then work out the integral of each ( using table above ) :
= electronic warfare − 3w + C
Sum, Difference, Constant Multiplication And Power Rules
Example: What is ∫(8z + 4z3 − 6z2) dz ?
Use the Sum and Difference Rule :
∫ ( 8z + 4z3 − 6z2 ) dz =∫8z dz + ∫4z3 dz − ∫6z2 dz
constant multiplication :
= 8∫z dz + 4∫z3 dz − 6∫z2 dz
exponent rule :
= 8z2/2 + 4z4/4 − 6z3/3 + C
simplify :
= 4z2 + z4 − 2z3 + C
Integration by Parts
See integration by Parts
Substitution Rule
See consolidation by Substitution
Read more: Is It Hard to Learn Computer Programming?
Final Advice
- Get plenty of practice
- Don’t forget the dx (or dz, etc)
- Don’t forget the + C
6834, 6835, 6836, 6837, 6838, 6839, 6840, 6841, 6842, 6843
