Integration by Substitution – Definition, Formula

 Substitution integration is an important integration method used when the functions to be integrated are complex functions or when direct integration of functions is not possible. The integral of a function is simplified by this integral method by permutation by reducing a given function to a simplified function. 

Integral by substitution is used when the integral of a given function cannot be obtained directly because the given algebraic function is not in canonical form. In addition, the given function can be reduced to the canonical form with proper substitution. Consider the indefinite integral of the functions f (x), ∫f (x). dx. Now this integral can be converted to another form by replacing x with g (t) and x = g (t). 

 I = ∫f (x). dx 

 Since x = g (t), dx / dt = g` (t) dx = g'(t). dt 

 I = ∫f (x). dx = ∫f (g (t)). g ′ (t). dt

Definite Integral: 

The definite integral is the area below the curve between the two fixed limits. For the function f (x) defined for the x-axis, the definite integral is expressed as abf(x)dx. Where a is the lower limit and b is the upper limit. Divide the area into rectangles and add them together to find the area below the curve between the two boundaries. The area will be more realistic if there are more rectangles. Therefore, divide the area into an infinite number of rectangles, each of the same (very small) size, and sum all the areas. This is the basic theory behind definite integrals. 

The integral is the sum of the areas, and the definite integral is used to find the area within the boundary. Integration was first studied in the 3rd century BC. Used to find areas of circles, parabolas, and ellipses. Learn more about definite integrals and the properties of integrals.

Formula of Definite integral: 

Given a continuous function f (x) in the interval [a, b], divide the interval into n subintervals of equal width Δx and select points from each interval x ∗ i. In that case, the definite integral of (x) goes from a to b is abf(x)dx = i=0nf(xi)Δx . 

Evaluating definite integrals in this way can be very tedious due to the complexity of the calculation. In the second half of this chapter, we will develop a method for calculating definite integrals without taking the range of Riemann sum. However, for now, you can rely on the fact that definite integrals represent the area under the curve, and you can use geometric formulas to evaluate definite integrals and calculate their area. This is done to make sure that a particular integral represents a region. Therefore, we will explain what to do if the curve of the function falls below the x-axis.

Integrals of Exponential Functions: 

The format of the exponential function is f (x) = Cex for the constant C and the linear shift, reciprocal, and quotient of such functions. Exponential functions are common in science, so it would be very useful to be able to integrate them. 

The exponential function is probably the most efficient function from an arithmetic point of view. The exponential function y = e^x is its own derivative and integral.

The following formula can be used to combine exponential functions:

∫ex dx=ex+C

∫axdx=ax/lna+C

Steps to Integration by Substitution: 

The following steps will help you perform this integration method with replacement. 

 Step 1: Select the new variable t for the particular function you want to reduce. 

 Step 2: Find the dx value of the given integral. Where f (x) is integrated with respect to x. 

 Step 3: Make the necessary permutations with the function f (x) and the new value dx. 

 Step 4: Consolidate the functions obtained after the replacement. 

 Step 5: Reset the initial variable x to get the final result.

Important Substitutions in Integration by Substitution: 

Below are some of the important replacements that help simplify the specified expression and make the integration process easier to perform. Let’s review the specific substitutions below for integration by substitution. 

 For the integral function f (√a2-x2), use x = aSinθ or x = aCosθ. Use x = aSecθ or x = aCosecθ to integrate the function f (√x2-a2). For the integral of the function f (x2 + a2). Use f (√x2 + a2) 

 x = aTanθ, or x = aCotθ. Use x = cos2θ to integrate the functions f (√a−xa + x) and f (√a + xa−x).

Difference between the definite integral and the indefinite integral: 

The definite integral f(x) is a number that defines the area under the curves within the specified limits. It has an upper limit and lower limit and it gives a definite answer. Whereas the indefinite integral f(x) is a function and it has no upper and lower limits. It gives a solution to the question “what function produces f(x) when it is differentiated?”.

CONCLUSION

The integrals listed here are called definite integrals and can be interpreted as signed regions of the area in the plane surrounded by the graph of the definite integral between two points on the real line. Traditionally, the area above the horizontal axis of the plane is positive and the area below is negative. Integral is also related to the concept of indefinite integral. The derivative of this function is a given function. In this case, they are called indefinite integrals. The Fundamental Theorem of Calculus provides a way to combine definite and derivative to calculate the definite integral of a function when the indefinite integral is known.