Antiderivatives are widely used in calculus to find the area under the curve. It is usually used to find the integrals of the functions with respect to their integrating variables. The function can be exponential, linear, logarithmic, polynomial, or constant.
Limits play a vital role in the antiderivatives to find the result of a definite type of antiderivative. In this post, we’ll learn the definition of antiderivative, its types, and formulas along with a lot of examples.
What is antiderivative (Integral)?
In calculus, an antiderivative is a function that reverses what the derivative does. One function has many antiderivatives but they all are written in the form of a function including the constant of integration.
Antiderivatives are also known as integral. But mostly it is related to the indefinite integral or it is the key part of the indefinite integral. In general, the antiderivative is the opposite of the derivative. There are two main types of integral (antiderivative).
- Definite
- Indefinite
The definite type of integral is frequently used in calculus. This type of integration has upper and lower bounds of the function like interval (r, s). The first term of the interval is the lower limit and the second value of the interval is the upper limit.
The formula of this type of integral is:
rs f(z) dz = F(s) – F(r) = L
- r & s are the interval values of the function
- f(z) is the integrand function.
- dz is the integrating variable of the function.
- F(s) – F(r) is the fundamental theorem of calculus to find the numerical value of the function by placing the upper and lower limit values.
- L is the result of the definite integral after applying the limit.
The other type of integral is indefinite integral. This type of integral does not have the upper and lower boundaries of the function. It is widely used to find the area under the curve and the volume. It has the symbol “ʃ”.
ʃ f(z) dz = F(z) + c
- f(z) is the integrand function.
- dz is the integrating variable of the function.
- F(z) is the result of the function after integrating the function.
- C is the integral constant.
How to find the antiderivative?
By using the rules & types of antiderivative the problems can be solved easily. Below are some examples of antiderivatives. You can also use an antiderivative calculator to calculate the antiderivative of a function with respect to its variable.
Example 1: For the definite integral
Integrate 5z4 + 2cos(z) – 6x2z5 – z3 + 12 with respect to z, having boundary values [3, 5].
Solution
Step 1: Take the given function and apply the integral notation along with the upper limit, lower limit, and integrating variable.
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz
Step 2: Apply the sum, difference, and constant rules of antiderivatives and write the definite integral notation separately to each function.
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 35 [5z4] dz + 35 [2cos(z)] dz – 35 [6x2z5] dz – 35 [z3] + 35 [12] dz
Step 3: Now apply the constant function rule of antiderivative and write the constants outside the integral notation.
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 535 [z4] dz + 2 35 [cos(z)] dz – 6x235 [z5] dz – 35 [z3] + 35 [12] dz
Step 4: Use the power, constant & trigonometric rules of antiderivative and integrate the above expression.
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 5 [z4+1 / 4 + 1]51 + 2 [sin(z)]51 – 6x2 [z5+1 / 5 + 1]51 – [z3+1 / 3 + 1]51 + [12x]51
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 5 [z5 / 5]53 + 2 [sin(z)]53 – 6x2 [z6 / 6]53 – [z4 / 5]53 + [12z]53
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 5/5 [z5]53 + 2 [sin(z)]53 – 6x2/6 [z6]53 – 1/5 [z4]53 + [12z]53
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = [z5]53 + 2 [sin(z)]53 – x2 [z6]53 – 1/5 [z4]53 + 12[z]53
Step 5: Now apply the fundamental theorem “rs f(z) dz = F(s) – F(r)” to get the result.
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = [55 – 35] + 2 [sin (5) – sin (3)] – x2 [56 – 36] – 1/5 [54 – 34] + 12[5 – 3]
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = [3125 – 243] + 2 [sin (5) – sin (3)] – x2 [15625 – 729] – 1/5 [625 – 81] + 12[5 – 3]
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = [2882] + 2 [sin (5) – sin (3)] – x2 [14896] – 1/5 [544] + 12[2]
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 2882 + 2sin (5) – 2sin (3) – 14896x2 – 544/5 + 24
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 2882 + 2sin (5) – 2sin (3) – 14896x2 – 108.8 + 24
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 2773.2 + 2sin (5) – 2sin (3) – 14896x2 + 24
35 [5z4 + 2cos(z) – 6x2z5 – z3 + 12] dz = 2797.2 + 2sin (5) – 2sin (3) – 14896x2
Example 2: For indefinite integral
Integrate 4z3 – 12sin(z) – 15z4 + 17z5 + 12z with respect to z.
Solution
Step 1: Take the given function and apply the integral notation along with the upper limit, lower limit, and integrating variable.
ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz
Step 2: Apply the sum, difference and constant rules of antiderivatives and write the definite integral notation separately to each function.
ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz = ʃ [4z3] dz – ʃ [12sin(z)] dz – ʃ [15z4] dz + ʃ [17z5] dz + ʃ [12z] dz
Step 3: Now apply he constant function rule of antiderivative and write the constants outside the integral notation.
ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz = 4 ʃ [z3] dz – 12 ʃ [sin(z)] dz – 15 ʃ [z4] dz + 17 ʃ [z5] dz + 12 ʃ [z] dz
Step 4: Use the power & trigonometric rules of antiderivative and integrate the above expression.
ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz = 4 [z3+1 / 3 + 1] – 12 [-cos(z)] – 15 [z4+1 / 4 + 1] + 17 [z5+1 / 5 + 1] + 12 [z1+1 / 1 + 1] + C
ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz = 4 [z4 / 4] – 12 [-cos(z)] – 15 [z5 / 5] + 17 [z6 / 6] + 12 [z2 / 2] + C
ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz = 4/4 [z4] – 12 [-cos(z)] – 15/5 [z5] + 17/6 [z6] + 12/2 [z2] + C
ʃ [4z3 – 12sin(z) – 15z4 + 17z5 + 12z] dz = z4 + 12cos(z) – 3z5 + 17z6/6 + 6z2 + C
you can also use an integral calculator to find the step by step solution of the problems. Follow the below steps to integrate the function using a calculator.
Step 1: Select the type of integral.
Step 2: Input the function.
Step 3: Select the integrating variable.
Step 4: Press the calculate button.
Step 5: The result with steps will show below the calculate button.
Summary
In this post, we have learned complete basics of antiderivatives. By reading the above post now you can grab all the initial concepts of integrals. You can easily solve the definite or indefinite integral problems by using the formulas and examples given in this post.
