Definite integral properties: Linearity , Reversal of limits, Zero width, Additivity, Bounds Inclusion, Comparison property, Absolute Value, Even and odd functions, Integration of a constant, Average value of a function, Additivity with zero contribution.

What is a definite integral?

A definite integral is a mathematical process that estimates the accumulation of quantities over an interval of time. The mathematical expression is as follows:

Here, and are the integration limits, is the function under consideration and represents an infinitesimally small change in. This process produces a number, which is commonly used to represent an area, volume, or other accumulated value.

Mathematical Representation

The definite integral of a function f(x) over the interval [a,b] is written as:

Here

• a: The lower limit of integration.
• b: The upper limit of integration.
• f(x): The function to be integrated.
• dx represents an infinitesimal change in $x$.

The Geometric interpretation

One of the simplest and most straightforward ways to understand definite integrals is through geometric interpretation. Consider a curve defined by on a graph. The definite integral from to calculates the net area between the curve and the -axis over the interval.

• Area above the x-axis: Contributions from add to the total.
• Area above the y-axis: contributions from subtracting from the total.

Properties of definite integrals

1.  Linearity: Definite integral respects the principles of addition and scalar multiplication.

This means you can split the integral across sums or factor out constants.

2. Reversal of limits : Reversing the order of the limits of integration modifies the sign of the integral.

3. Zero width: The outcome is 0 if the integral’s upper and lower limits are equal.

This is because no area is accumulated over a zero-width interval.

4. Additivity (splitting intervals): A definite integral over an interval can be split into subintervals.

This is useful for breaking complex integrals into manageable parts.

5. Bounds Inclusion: If f(x) is continuous on [a,b], then:

This property is often used to handle piecewise functions.

6. Comparison property: If f(x) ≤ g(x) for any x in  [a,b], then:

If the functions are strictly ordered, the inequality becomes strict.

7. Absolute Value: The absolute value of an integral does not equal the integral of the absolute value in general.

This inequality reflects that the definite integral accounts for signed areas.

8. Even and odd functions

• For an even function F(x)=F(-x) , integrated over symmetric limits

• For an odd function F(x)=-F(-x), integrated over symmetric limits

9. Integration of a constant: If f(x) = c , where c is a constant, then:

This is the area of a rectangle whose width is b−a and its height is c.

10. Average value of a function: The definite integral can be used to compute the average value of f(x) over [a,b]

11. Additivity with zero contribution: If $f(x)=0$ over an interval, then:

These are Definite integral properties.

Example 1:

Evaluate

Using linearity:

Compute each term

S0

Example 2:

Evaluate

Step 1: Analyze the problem

The integrand suggests that a substitution related to the square root will simplify the integral . Let’s proceed with a trigonometric substitution.

Step 2: Sustitution

Let,

x=Sec(θ) , dx=Sec(θ) tan(θ) dθ 

The integral becomes

Simplify

Thus the integrand simplifies to

Step 3: Adjust limits of integration

When x=1 :

x= sec(θ) ⟹sec(θ)=1 ⟹θ=0

when x=4

x= sec(θ) ⟹sec(θ)=4 ⟹θ=sec−1(4)

The new limits are

Step 4: Integrate

The integral is now

The antiderivative of sec⁡^2(θ) is $tan(\theta ).$

Step 5: Evaluate

Substitute the limits

and  at θ=0 tan(0)=0

Thus

Final Answer

Learn more indefinite integral

Solve Indefinite integral :The indefinite integral of a function  f(x) is a family of functions  F(x) such that the derivative of  F(x) equals  f(x). In other words:

The indefinite integral is written as:

Where

• F(x) is the integrand ( the function being integrated)
• dx indicates the variable of integration.
• F(x) is the antiderivative of f(x).
• The family of all potential antiderivatives is represented by the constant of integration, C.

Solve Indefinite integral math problems

Example 1

Step 1: Expand sin3x using the angle formula

We are aware of the expansion for sin(3x) ⁡.

Substitute this in the integral

Expand the integral

Step 2: Solve the first term

For 3∫cos^4(x)sin(x)dx, use substitution: Let u=cos(x), so du=−sin⁡(x)dx This becomes:

Substitute back u=cosx

Step 3: Solve the second term

For −4∫cos^⁡4(x)sin⁡3(x) dx, use

So the integral becomes

Substitute u=cos⁡(x), du=−sin⁡(x)dx:

Simplify

Integrate

Substitute back u=cos(x)

Step 4: Combine results

Add the two terms together

Simplify

Final answer

Example 2

Step 1: Use the half-angle identity

We know the identity

Substitute this into the integral

Simplify the square root

Factor out √2

Step 2: Substitution

Let u=x/2, so x=2u and dx=2 du. Substitute:

Simplify

Step 3: Integrate

The integral of cos (u) is sin(u)

Step 4: Back substitute

Recall that u=x/2​, so:

Final answer

Example 3

Step 1: Use the product to sum identities

The product of two cosines can be expressed as

Apply this identity to cos2xcos3x:

Since cos⁡(−𝑥) = cos⁡𝑥 this can be simplified to:

Step 2: Substitute into the original integral

Now substitute this result into $\int \cos x\cos 2x\cos 3xdx$:

Simplify

Step 3: Solve term separately.

a) First term: ∫cos⁡xcos⁡5x dx

Use the product-to-sum identity again:

This becomes: as cos(−4x)=cos4x

Substitute into the integral:

Evaluate

Thus

b) Second term: ∫cos²x dx

Use the identity cos²x= (1+cos2x)/2

Evaluate

Thus

Step 4: Combine the results.

Combine the outcomes for the two terms now:

Simplify

Final answer

With practice, you’ll discover that solving integrals is a natural and gratifying process! Here are some basic integral rules

Some practice problems

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Partial fraction formula: Integration by Partial Fractions is a method for integrating rational functions that are fractions of polynomials. The goal is to reduce a complex rational function to simpler fractions that are easier to integrate.

Steps for Partial Fraction Integration:

1.  Check to see if the degree of the numerator is less than that of the denominator:

If the degree of the numerator is greater than or equal to the degree of the denominator, use long division to rewrite the integral so that the degree of the numerator is less than that of the denominator.
2. Factor the denominator, if possible:

Try to break the denominator down into irreducible elements (linear or quadratic).
3. Set up the partial fractions: Depending on the factorization of the denominator, break the rational function into simpler fractions:

• For linear factors (x-a) , use the form

• Use the form for repeated linear components  (x−a)^n

• For quadratic factors that cannot be reduced (x^2+bx+c) use the form

• Use the following expression for repeated irreducible quadratic factors: (x^2 + bx + c)^n

4. Solve for the constants: After setting up the partial fractions , multiply both sides of the equation by the common denominator and equate coefficients of like powers of x to solve for the constants.

5. Integrate the simpler fractions: After determining the constants, integrate each term separately.

Partial fraction formula Partial fraction formula

1. Linear factors in the denominator

If the denominator consists of different linear factors:

Example:

Steps

1. Decompose

2. Solve for A and B.

3. Integrate each term.

Solution: A=1, B=1

2. Repeated Linear Factors

If the denominator includes repeated linear factors:

Example:

Steps

1. Decompose

2. Integrate each term.

Solution:

3. Quadratic Factors

If the denominator includes an irreducible quadratic factor:

Example:

1. For irreducible x^2+4, rewrite:

2. Solve for coefficients if needed (A = 0, B= 1)

3. Integrate using

Solution:

4. Repeated Quadratic Factors

For several irreducible quadratic factors:

Example:

Steps

1. Decompose

2. Solve for coefficients.

3. Integrate each term.

Solution:

5. Mixed Linear and Quadratic Factors

When the denominator contains both linear and quadratic factors:

Example:

Steps

1. Decompose

2. Solve for A, B, and C.

3. Integrate each term separately.

Example :

Step 1 : First, configure the partial fractions.

As (x+1)^2 (x−2), the denominator is already factored as (x+1)^2 (x−2). The integrand will therefore be represented as a sum of partial fractions of the following form:

Step 2: Multiply both sides by (𝑥+1)^2 ( 𝑥 − 2 ) 

Step 3: Expand the right side.

Next, expand each phrase on the right-hand side:

Thus we get

Simplifying

Step 4: Compare the coefficients.

We equate the coefficients of powers of 𝑥 on both sides of the equation.

• Coefficient of x^2: A+C=2
• Coefficient of $x$: A+B+2C=0
• Constant term: −2A−2B+C=−1

Step 5: solve the system of equations.

We will now solve the system of equations:

1. A+C=2
2. -A +B+2C=0
3. -2A-2B+2C=-1

From equation 1

C=2-A

Substitute C=2−A into equations (2) and (3):

From equation 2

From equation 3

Now solve equations (4) and (5).

Based on equation (4):

𝐵 = 3 𝐴 − 4

Substitute B=3A−4 into equation (5):

Now substitute A=11/9 inti C = 2-A

Finally , substitute A = 11/9 into B = 3A-4 :

Step 6: Write a partial fraction decomposition.

We currently have:

Step 7: Integrate the terms.

We may now integrate each phrase separately.

Thus, the solution is:

This is the final answer.

Practice problems

Solve the integral using partial fraction

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