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# Integration Rules

## 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.

### 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 .

### 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

### 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

### 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

### 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

### 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

• 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

reference : https://thefartiste.com
Category : Tech 