Question

What is ∫x3 dx ? ................

Answer 1

your solution is attached above .....

Answer 2

Integration

Integration can be used to find areas, volumes, central points and many useful things. But it is often used to find the area underneath the graph of a function like this:

 integral area

 

The integral of many functions are well known, and there are useful rules to work out the integral of more complicated functions, many of which are shown here.

There are examples below to help you.

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+1n+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 table above it is listed as being −cos(x) + C

It is written as:

∫sin(x) dx = −cos(x) + C

Power Rule

Example: What is ∫x3 dx ?

The question is asking "what is the integral of x3 ?"

We can use the Power Rule, where n=3:

∫xn dx =  xn+1n+1  + C

∫x3 dx =  x44  + C

Example: What is ∫√x dx ?

√x is also x0.5

We can use the Power Rule, where n=½:

∫xn dx =  xn+1n+1  + C

∫x0.5 dx =  x1.51.5  + C

Multiplication by constant

Example: What is ∫6x2 dx ?

We can move the 6 outside the integral:

∫6x2 dx = 6∫x2 dx

And now use the Power Rule on x2:

= 6  x33  + C

Simplify:

= 2x3 + C