Learn meaning of antiderivative and graphical meaning of integral calculus formulas. Practice problems for calculating antiderivative of x^n, trigonometric, exponential and average value of function.

- A function \(F(x)\) is an antiderivative of the function \(f(x)\),

if \(\dfrac{d}{dx} F(x) = f(x)\)

- To indicate \(F(x)\) as an antiderivative of \(f(x)\), the following notation is used

\(F(x) = \int f(x) dx\)

- In General :

As \(\dfrac{d}{dx} [F(x) +C]\;=f(x)\)

so, \(\int f(x) dx= F(x) +C\)

where C is a constant of integration.

- Antiderivatives are also known as indefinite integrals.

A \(g^2(x)\)

B \(g(x)\)

C \(f(x)\)

D \(\dfrac{f(x)}{2}\)

**Case 1 :**

- Antiderivative of \(x^n\) when \(n \neq -1\)
- Let \(f(x) = x^n\)
- Antiderivative of \(f(x)= \int f(x) dx\)

= \(\int x^n dx\)

\(= \dfrac{x^{n+1}}{n+1} +C\) where \(n \neq -1\)

**Case 2 :**

- Antiderivative of
**\(x^n\)**when n= 1 - Let \(g(x)=x^{-1}\)
- Antiderivative of \(g(x) = \int g(x) dx\)

\(= \int x^{-1} dx\)

\(= \displaystyle \int \dfrac{1}{x} dx\)

\(= log _e x+C\)

A \(\displaystyle \int x^5 dx = \dfrac{x^6}{6} + C\)

B \(\displaystyle \int x^{-1} dx = log_e x + C\)

C \( x^{-3} dx = \dfrac{-1}{2x^2} + C\)

D \(\displaystyle \int x^{10} dx = 10\,log_e x+ C\)

- Since, velocity is the derivative of displacement so, displacement is the antiderivative of velocity \(v\).
- Considering velocity as a function of time and taking very small interval of time, then the summation of product of these small intervals with velocity gives the displacement.
- For the calculation of displacement between time interval t
_{1}to t_{2}, the summation is denoted as

\(\displaystyle\int\limits_{t_1}^{t_2} v . dt = \)Displacement

- In general

Let \(\dfrac{d}{dx} F(x) = f(x)\)

then, antiderivative \(\int f(x)\, dx = F(x) +C\)

- Performing this integration for the limits a to b

\(\displaystyle\int\limits_{a}^{b} f(x) dx = [F(x) + C]^b_a\)

\(= [F(b) + C] - [F(a) + C]\)

\(\displaystyle\int\limits_{a}^{b} f(x)dx = [F(b)-F(a) ]\)

**Conclusion : \(\displaystyle\int\limits_{a}^{b} f(x) dx\)** represents summation of small units into the end points [a,b],

where **a** is the lower limit

**b** is the upper limit

- Consider a function \(y= f(x) \), as shown in figure.

- The area under the curve is given by the definition of definite integral.

\(A = \displaystyle \int\limits^b_a f(x) dx\)

- Average value of the function can be calculated as

Area of PQRS \(\cong\) Area under the curve

\(f_{avg} × (b-a) = \displaystyle\int\limits ^b_a f(x) dx\)

\( f_{avg} = \dfrac{\displaystyle \int \limits^b_af(x)dx}{(b-a)}\)

A 36

B 28

C 42

D 18

**(1) ** If \(f(x) = sin(x)\)

\(\int f(x) \,dx = \int sin x \,dx = - cos \,x +C\)

**(2)** If \(f(x) = cos(x)\)

\(\int f(x)\, dx = \int cos \,x \,dx = sin \,x +C\)

**(3) ** If \(f(x) =tan(x)\)

\(\int f(x)\, dx = \int tan \,x \,dx = \ell n |sec \,x| +C\)

**(4) ** If \(f(x) =cot(x)\)

\(\int f(x) \,dx = \int cot\,x \,dx = \ell n |sin \,x| +C\)

**(5) ** If \(f(x) =sec(x)\)

\(\int f(x) \,dx = \int sec \,x \,dx = \ell n |sec \,x + tan\,x| +C\)

**(6)** If \(f(x) =cosec(x)\)

\(\int f(x) \,dx = \int cosec \,x \,dx = \ell n |cosec \,x + cot \,x| +C\)

A \(\displaystyle \int tan\,5x \,dx = \dfrac{1}{5} \ell n |sec \,5x| +C\)

B \(\displaystyle \int cosec\,2x \,dx = \dfrac{1}{2} \ell n |cosec\,2x \,-cot\,2x| +C\)

C \(\displaystyle \int cos\,3x \,dx = \dfrac{1}{3} |sin \,3x| +C\)

D \(\displaystyle \int sec\,3x \,dx = 3\, |-cosec \,x + tan \,x| +C\)

- Consider an exponential function \(f(x) = e^x\)
- Antiderivative of \(f(x) = \int f(x) dx\)

\(= \int e^x \,dx\)

\(= e^x + C\)

A \(e^x +C\)

B \(e^{2x} +C\)

C \(e^{4x} +C\)

D \(\dfrac{1}{2}e^{2x} +C\)

- Consider a continuous function \(y = f(x)\) defined between limits \(x=a\) to \(x =b\), as shown in figure with a graph of any arbitrary shape.
- The graph is bounded by curve \(y = f(x)\), \(y= 0\) and the lines \(x=0\) and \(x=b\).

- Consider a strip of thickness \(dx\) at a distance \(x\) from the origin with length parallel to \(y{-axis}\), as shown in figure.
- Area of the shaded region is

\(dA = f (x)\, dx\)

- If the area is divided into 'n' strip of equal width, then complete area of the region below the graph and above the \(x \) - axis confined between the coordinates at \(x=a\) to \(x=b\) can be obtained by summing up the area of 'n' individual strip.
- For better approximation, 'n' is considered to be very large.
- This summation is represented by definite integral.

\( A = \displaystyle\int\limits^b_a f(x) dx\)