Integration help worksheet question – free integration worksheet

Integration Made Easy

By Masood Amir

Calculus:

Calculus is a tool used to measure change or variation of a function with respect to the independent variable.

Differential Calculus:

It is the branch of calculus used to measure change or variation of a function in a very small interval of time, the techniques use to measure such changes is called “differentiation”.

Integral Calculus:

It is the branch of calculus used to measure changes or variation over an interval of independent variable, e.g to find length of curve, the area of region and the volume of a solid in a specified period of time.

The technique used to measure such changes or variation is called “Integration” or “Antiderivatives”. It a reverse process of differentiation.

Mathematically, Integration is defined as “ If f’(x) represents the differential coefficient of f(x), then the problem of integration is given f’(x), find f(x) or given dy/dx, find y.

Notation:  ”∫ ” is used to show the integration, it is a symbol of “S” derived from the word “Sum”. i.e. Integration is a process in which we have to sum up the derivatives over a specified interval and to find the function.

Techniques of Integration:

As we know that integration is the reverse process of differentiation, our problem is to find the function f(x) or Y, when f’(X) or dy/dx is given.

dy/dx = f’(X)

∫dy = ∫f ’(X)dx

Y= f(x) is our solution

Ist Formula of Integration (1st Rule of Integration)

Indefinite Integration:

Ist Formula of Integration (Ist Rule of Integration):

 If  y = xⁿ

∫ dy = ∫ xⁿdx

y = xⁿ⁺¹/n+1  + C

Why “C”:

In the process of differentiation, we eliminate constant, as the derivative of a constant is “zero”.

So, In functions like Xⁿ, Xⁿ+ 6, Xⁿ + 3 ,  Xⁿ – K, the derivatives of all of them is X^n-1, in finding the anti derivative of  X^n-1, we put a constant “C”, as we don’t know which constant was present in the original function, and this can be found If we have initial boundary values (Definite Integral).

Example: Solve ∫x³dx

Solution:        x³⁺¹/3+1 + C   = x⁴/4 + C

Example: Solve ∫(x³ + x² + 5x + 6)dx

Solution:      ∫x³dx + ∫x²dx + ∫5xdx + ∫6dx

                         X⁴/4 + x³/3 + 5x²/2+ 6x + C

 

Worksheet # 1

 

Find the Integral of the following:

 

SET 1

a. ∫(x³-4x²+5x-6)dx                       

1. ∫(3x⁵-4x³+3x²)dx

c. ∫(ax⁵-bx⁴)dx                               

1. ∫(^x3/2-5x^4/3+3x²)dx

e.∫(⁴√3x² -2x)dx                     

1. ∫(3x + 5x² –x³/2-0.4x⁴)dx

g. ∫(x(8x-1/2)dx                                       

1. ∫(2-x)(4+3x)dx
2. ∫(x^-3 + x^-4)dx                               
3. ∫(2x³-3)3x⁴dx

k. ∫(4x⁷ + 3x¹² -5x⁸ + 2x -1)dx            

1. ∫(ax³ – bx² + cx –d)dx
2. ∫(1/x³ + 2/x² -6)dx
3. ∫(-3x^-8 + 2√x)dx
4. ∫((x³ -5)(2x + 5)dx
5. ∫(7x^-6 + 5√x)dx

More Questions

Evaluate ∫ (2x + 3) dx.

Answer: x^2 + 3x + C,

Find ∫ (5e^x + 2x) dx.

Answer: 5e^x + x^2 + C,

Calculate ∫ (3sin(x) + 4cos(x)) dx.

Answer: -3cos(x) + 4sin(x) + C,

Evaluate ∫ (4x^3 – 2x^2 + 5) dx.

Answer: x^4/4 – 2x^3/3 + 5x + C,

Find ∫ (6/x) dx.

Answer: 6ln|x| + C,

Calculate ∫ (2sec^2(x)) dx.

Answer: 2tan(x) + C,

Evaluate ∫ (3e^(2x) + 2/x) dx.

Answer: (3/2)e^(2x) + 2ln|x| + C,

Find ∫ (4sinh(x) + 2cosh(x)) dx.

Answer: 4cosh(x) + 2sinh(x) + C,

Calculate ∫ (2x^2 – 3x + 1) dx.

Answer: (2/3)x^3 – (3/2)x^2 + x + C,

Evaluate ∫ (7x^2 + 6x + 5) dx.

Answer: (7/3)x^3 + 3x^2 + 5x + C,

Find ∫ (2sec(x)tan(x)) dx.

Answer: 2sec(x) + C,

Calculate ∫ (e^x + 1/x^2) dx.

Answer: e^x – 1/x + C, where C is the constant of integration.

Evaluate ∫ (2cos(3x) – 3sin(2x)) dx.

Answer: (2/3)sin(3x) + (3/2)cos(2x) + C, where C is the constant of integration.

Find ∫ (5x^4 + 4x^3 + 3x^2) dx.

Answer: (5/5)x^5 + (4/4)x^4 + (3/3)x^3 + C, where C is the constant of integration.

Calculate ∫ (2/(x+1)) dx.

Answer: 2ln|x+1| + C, where C is the constant of integration.

Evaluate ∫ (6e^(2x) + 2/x^3) dx.

Answer: (3e^(2x) – 1/x^2) + C, where C is the constant of integration.

Find ∫ (3cosec^2(x)cot(x)) dx.

Answer: -3cosec(x) + C,

Calculate ∫ (x^2 + √x + 1) dx.

Answer: (1/3)x^3 + (2/3)x^(3/2) + x + C,

Evaluate ∫ (4sin(3x) + 3cos(4x)) dx.

Answer: -(4/3)cos(3x) + (3/4)sin(4x) + C,

Find ∫ (5x^3 – 2x^2 + 1/x) dx.

Answer: (5/4)x^4 – (2/3)x^3 + ln|x| + C,

Calculate ∫ (2/(x^2 + 4x + 5)) dx.

Answer: 2arctan(x + 2) + C,

Evaluate ∫ (e^x – e^(-x)) dx.

Answer: e^x + e^(-x) + C,

Find ∫ (2x^3 – 3x^2 + 4x – 5) dx.

Answer: (2/4)x^4 – (3/3)x^3 + 2x^2 – 5x + C,

Calculate ∫ (3/x^2) dx.

Answer: -3/x + C,

Evaluate ∫ (6sin(x)cos(x)) dx.

Answer: 3sin^2(x) + C,

Find ∫ (x^2 + 2x + 3) dx.

Answer: (1/3)x^3 + x^2 + 3x + C,

Calculate ∫ (3e^(2x) – 2e^(-x)) dx.

Answer: (3/2)e^(2x) + 2e^(-x) + C,

Evaluate ∫ (4x + 5) dx.

Answer: 2x^2 + 5x + C,

Find ∫ (6tan(x)sec^2(x)) dx.

Answer: 6tan(x) + C,

Calculate ∫ (1/(x^2 + 9)) dx.

Answer: (1/3)arctan(x/3) + C,

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