Question

How to solve problems using integrals? Example

Answer 1

So Integral and Derivative are opposites.

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.
Common Functions Function Integral Rules 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
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

Example :From the table 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 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 also x^0.5

We can use the Power Rule, where n=½:

∫xⁿ dx = xⁿ⁺¹/n+1+ C

∫x^0.5 dx = x^1.5/1.5+ C

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 x³/ 3 + C

Simplify:

= 2x³ + C