 # Differentiation and Integration Formula

Differentiation and Integration Formulas have many important formulas. You’ll read about the formulas as well as its definition with an explanation in this article.

## Definition of Differentiation

A derivative of a function related to the independent variable is called Differentiation and it is used to measure the per unit change in function in the independent variable. If y = f(x) is a function of x, then the rate of change of y per unit change in x is given as:

 dy/dx

## Important Differentiation Formulas

 f(a) = sin a, so f'(a) = cos a f(a) = cos a, so f'(a) = -sin a f(a) = tan a, so f'(a) = sec2a

## Definition of Integration

The integration of a function f(a) is given by F(a) and is written as

$$\begin{array}{l}\int\end{array}$$
f(a) da = F(a) +C, where Right hand side shows integral of f(a) with respect to a

F(a) is called primitive, f(a) is called the integrand and C is constant of integration, a is variable.

## Important Integration Formulas

 $$\begin{array}{l}\int\end{array}$$ sin a da = -cos a + c $$\begin{array}{l}\int\end{array}$$ cos a da = sin a + c $$\begin{array}{l}\int\end{array}$$ sec2 a da = tan a + c

## Solved Examples

Example 1: Find the derivative of x2cos x.

Solution:

Let f(x) = x2 cos x

Differentiate the given function with respect to x,

f’(x) = d/dx (x2 cos x)

= x2 d/dx(cos x) + cos x d/dx(x2)

= x2 (-sin x) + cos x (2x)

= 2x cos x -x2 sin x

Example 2: Find the derivative of f(x) = x + sin x.

Solution:

Given function is f(x) = x + sin x

Differentiate with respect to x,

f’(x) = d/dx(x + sin x)

= d/dx(x) + d/dx (sin x)

= 1 + cos x

Example 3: Evaluate: ∫7 cos x dx

Solution:

∫7 cos x dx

= 7 ∫cos x dx

= 7 sin x + C

Example 4: Evaluate:  ∫6 sin 3x dx

Solution:

∫6 sin 3x dx

= 6  ∫ sin 3x dx …..(i)

Let u = 3x  …..(ii)

du = 3dx

dx = ⅓ du …..(iii)

From (i), (ii) and (iii),

= 6  ∫sin u (⅓) du

= (6/3)  ∫ sin u du

= 2 (-cos u) + C

Substituting u =  3x in the above step,

= -2 cos 3x + C