Class 12 Maths Integrals | Fundamental Theorem of Calculus |

__Fundamental Theorem of Calculus__

There are some fundamental theorems which are going to discuss here one by one.

**Area function**

Let x be the given point in [a, b] as shown in the figure.

Then _{a}ʃ^{b} f(x) dx represents the area of the shaded region. The function A(x) is represented as Area function and is calculated as

A(x) = _{a}ʃ^{b} f(x) dx

According to this definition, two fundamental theorems are stated as:

**First fundamental theorem of integral calculus**

**Theorem 1:** Let f be a continuous function on the closed interval [a, b] and let A (x) be the area function.

Then A′(x) = f (x), for all x ∈ [a, b].

**Second fundamental theorem of integral calculus**

**Theorem 2:** Let f be continuous function defined on the closed interval [a, b] and F be an anti derivative of f. Then _{a}ʃ^{b} f(x) dx = [F(x) _{a}]^{b} = F(b) – F(a), where F(x) is the indefinite integral of f(x).

**Problem: Evaluate the following definite integrals**

**(a) ʃ _{-1}^{1} (x + 1) dx (b) **

**Solution:**

(a) Let I = ʃ_{-1}^{1} (x + 1) dx

Now, ʃ (x + 1) dx = x^{2}/2 + x = F(x)

By second fundamental theorem of calculus, we get

I = F(1) - F(-1)

= (1/2 + 1) – (1/2 - 1)

= 1/2 + 1 – 1/2 + 1

= 2

So, ʃ_{-1}^{1} (x + 1) dx = 2

(b) Let I = ʃ_{2}^{1} dx/x

Now, ʃ dx/x = log|x| = F(x)

By second fundamental theorem of calculus, we get

I = F(3) - F(2)

= log|3| - log|2|

= log 3 – log 2

= log(3/2)

So, ʃ_{2}^{1} dx/x = log(3/2)

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