Integration Rules

Integration Rules

Integration

integral area
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 !
integral vs derivative
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 .

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

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

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Category : Tech

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