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# Math Examples of Comparison between Integration and Differentiation

• ## Differentiate and Integrate the function f(x) = x + 1?

First we will see the differentiation of the function f ( x ), the differentiation of function is shown below-
=>  d ( f ( x ) ) / dx = d ( x + 1 ) / dx,
=>  d ( f ( x ) ) / dx = d ( x ) / dx + d ( 1 ) / dx,
=> d ( f ( x ) ) / dx = 1 + 0,
=> d ( f ( x ) ) / dx = 1.

Therefore differentiation of function f ( x ) is equals to 1.
Now we will see Integration of function f ( x ), we get-
=> ∫ f ( x ) dx = ∫ ( x + 1 ) dx,
=> ∫ f ( x ) dx = ∫ x.dx  + ∫ 1.dx,
=> ∫ f ( x ) dx = x2 / 2 + x + c,

So integration of function f ( x ) is equals to = x2 / 2 + x + c, where ‘c’ is an arbitrary constant.

## Differentiate and integrate the function f ( x ) = x2 + x?

Differentiation of the function is given by-
=>  d ( f ( x ) ) / dx = d ( x+ x ) / dx,
=>  d ( f ( x ) ) / dx = d ( x2 ) / dx + d ( x ) / dx,
=>   d ( f ( x ) ) / dx = 2x + 1,

Therefore differentiation of function f ( x ) is equals to 2x + 1.
Integration of function is given below-
=>   ∫ f ( x ) . dx = ∫ ( x+ x ). dx,
=>  ∫ f ( x ) . dx =  ∫ ( x) .dx + ∫ ( x ) .dx,
=>  ∫ f ( x ) . dx = x/ 3 + x/ 2 + c,

Therefore Integration of function f ( x ) is equals to x/ 3 + x/ 2 + c, where ‘c’ is a constant.

## Differentiate and integrate the function f ( x ) = sin2 x ?

Differentiation of function is shown below-
=>  d ( f ( x ) ) / dx = d ( sin2 x ) / dx,
=>   d ( f ( x ) ) / dx = 2 sin x .d ( sin x ) / dx,
=>  d ( f ( x ) ) / dx = 2 sin x . cos x,
Therefore differentiation of function f ( x ) is equals to 2 sin x . cos x.
Integration of function f ( x ) can be done as shown below-
=>  ∫ sin x . dx = ∫ ( 1 – cos 2x ) / 2. Dx,
=>  ∫ sin x . dx = ∫ ½  dx - ∫ ( cos 2x ) / 2 .dx,
=>  ∫ sin x . dx = x / 2 – 1 / 4 sin 2x + c,
Therefore Integration of sin 2x is equals to x / 2 – 1 / 4 sin 2x + c.

## Differentiate and integrate the function x3 + x2?

Differentiation of function f ( x ) = x3 + x2 is shown below-
=>  d ( f ( x ) ) / dx = d ( x3 + x2 ) / dx,
=>   d ( f ( x ) ) / dx = d ( x3 ) / dx + d ( x2 ) / dx,
=>  d ( f ( x ) ) / dx = 3x+ 2x
Therefore differentiation of function f ( x ) is equals to 3x+ 2x.
Integration of function f ( x ) = x3 + x is shown below-
=>  ∫ ( x3 + x). dx = ∫ x.dx + ∫ x.dx,
=>  ∫ ( x3 + x). dx = x/ 4 + x/ 3 + c,
Therefore Integration of f ( x ) is equals to x/ 4 + x/ 3 + c.

## Differentiate and integrate the function f (x)= sin x + cos x?

Differentiation can be done as f ( x ) = sin x + cos x as shown below-
d ( f ( x ) ) / dx = d ( sin x + cos x ) / dx,
=>  d ( f ( x ) ) / dx = d ( sin x ) / dx + d ( cos x ) / dx,
=>  d ( f ( x ) ) / dx = cos x – sin x,

Therefore differentiation of function f ( x ) is equals to ( cos x – sin x ).
Integration of function f (x)  is shown below-

=>  ∫ f ( x ) dx = ∫ ( sin x + cos x ) dx,
=>  ∫ f ( x ) dx =  ∫ ( sin x ) dx +  ∫ ( cos x ) dx,
=>  ∫ f ( x ) dx = -cos x + sin x + c,
Therefore Integration of function f ( x ) is equals to ( -cos x + sin x + c ).
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