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U-substitution formula, in calculus, often known as integration by substitution, is a method for finding integrals. The u-substitution method is simply a technique to simplify integrals by substituting a function u = g(x) and its derivative du/dx = g'(x) in place of another function f(x). The anti-derivative is found using the fundamental theorem of calculus.
The formula of U-substitution is:
| ∫f(g(x)) g'(x) dx = ∫f(u) du |
Here, u = g(x) and du/dx = g'(x), which means that dx = du/g'(x).
By substituting these values into the integral, the same can be converted into a simpler integral in terms of u.
- The u-substitution method is useful when dealing with integrals that involve composite functions or when the integrand is a product of different functions.
- The u-substitution formula is an alternative method for using the chain rule of differentiation.
- The chain rule for differentiation is comparable to this u substitution formula.
- The supplied function is substituted by 'u' in the u-substitution formula, and then u is integrated using the fundamental integration formula.
- We substitute the real function in place of u after integration.
- The U-substitution rule appears to be the inverse of the chain rule.
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Key Terms: Calculus, Integration, Chain Rule of Differentiation, Composite Function, Differentiation, Definite Integral, Indefinite Integral, Fundamental theorem of Calculus, Substitution Method
What is Integration?
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The summing of discrete data is represented by integration.
- Integration is a process that involves finding the integral, or the area under the curve, of a function.
- It is the reverse process of differentiation and is typically used to determine the original function from its derivative.
- Integration is an essential concept in calculus which includes finding the integral or the antiderivative of a function.
There are two types of integration:
- Indefinite Integration: Indefinite integration involves finding the antiderivative of a function, which is the set of all possible functions whose derivative is the given function.
- Definite Integration: Definite integration involves finding the area under the curve of a function between two specified limits.
Integration by Substitution Detailed Video Explanation
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Integration by Substitution Method
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Integration by substitution, also known as u-substitution or change of variables, is a method of finding integrals which includes substituting a new variable in place of the existing variable in the integral.
- The new variable is typically chosen such that the integral simplifies, making it easier to evaluate.
- Any given integral is changed into a simple form of integral using this approach of integration by substitution by substituting other variables for the independent variable.
The general form of integration by substitution is:
| ∫f(g(x)).g'(x).dx = f(t).dt where t = g(x) |
When we make a substitute for a function whose derivative is also present in the integral, the method of integration by substitution is particularly beneficial. As a result, the function becomes simpler, and the fundamental integration formulae may be employed to integrate the function.
Integration by Substitution Example
For Example, let us integrate 2x sin (x² + 1) in terms of x.
As we know that the derivative of (x² +1) = 2x
Now, let us substitute x + 1= k so that 2x dx = dk. Hence,
∫∫2x sin (x²+1) dx = ∫∫sin k dk
= - Cos k + C
= - Cos (x² +1) + c
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What is U Substitution Formula?
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The main function is substituted by 'u' in the U substitution formula, and the variable u is then integrated using the fundamental integration formula, however after integration, we substitute the actual function in place of u. The U substitution formula is as follows:
| ∫f(g(x)) g'(x)dx = ∫f(u)du |
Where, u = g(x) and du/dx = g'(x)
This formula simply consists of making a substitution by letting u to be g(x) [i.e. u = g(x)], then replacing every occurrence of x with u and dx with du.
The derivative of u with respect to x, du/dx, can be determined using the chain rule: du/dx = g'(x).
How to Use U-Substitution Formula?
Here's a step-by-step guide on how to use the u-substitution formula:
- Step 1: Identify one part of the integrand which can easily be simplified by substitution. This can be a function occurring inside another function or under a radical.
- Step 2: Let “u” be the simplified part of the integrand and substitute it with u in the integrand.
- Step 3: Find the derivative of u with respect to x, du/dx, by taking the derivative of u with respect to x using the chain rule.
- Step 4: Replace all occurrences of the original variable x and its differential dx in the integrand with u and du, respectively.
- Step 5: Calculate the new integral in terms of u.
- Step 6: Replace back u in terms of x to get the final result.
U-Substitution Formula Example
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Indefinite Integrals
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Consider a derivative function f'(x) that is defined in an interval and is differentiable. We can use indefinite integrals, often known as antiderivatives, to reverse the differentiation process and determine the function F(x) whose derivative is given to us. For any given derivative, there are an infinite number of anti-derivatives. Integrals are represented mathematically by the symbol.
(x) + C = F(x)
Here, C stands for an arbitrary constant.
Indefinite Integrals Example
For Example, let us integrate ∫(2x+8) (x2+8x) 6dx ∫(2x+8) (x2+8x) 8dx using the substitution formula.
Let u = (x2 + 8x)
So that, du = (2x+8) dx.
Substitute the value of u and du in ∫ (2x+8) (x2+8x) 6dx ∫(2x+8) (x2+8x) 8dx, replacing all forms of x, getting
Using U Substitution Formula,
∫(2x+8) (x2+8x) 6dx = ∫(x2+8x) 6(2x+8)dx ∫(2x+8) (x2+8x) 8dx = ∫(x2+8x) 6(2x+8) dx
∫u8 du ∫ u8 du
= u77 + cu77 + c
= (x2+6x) 77+c(x2+6x) 77+c
Read More: Reduction Formula in Integration
Definite Integrals
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A definite integral is an integral wherein the integral limits, i.e., the upper and lower bounds of the integral, have been given. A definite integral represents the area between the curve of a function and the x-axis, within the given limits of integration.
The notation for a definite integral is:
→ ∫ab f(x) dx
where,
- "a" is the lower limit of integration
- "b" is the upper limit of integration,
- "f(x)" is the function to be integrated
- "dx" denotes that the variable of integration is "x".
The formula of Definite Integral is:
| \(\int_{a}^{b} f(x) d x=\lim _{n \rightarrow \infty} \sum_{r=1}^{n} h f(a+r h)\) |
Here, \(h=\frac{b-a}{n}\)
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Things to Remember
- The u-substitution method can be defined as a method used to integrate functions that can be converted into a simpler form using a substitution.
- A given integral f(x) dx can be changed into another form using the substitution method by changing the independent variable x to t. Substituting x = g accomplishes this (t).
- Substitution method is a technique for simplifying a system of equations by expressing one variable in terms of another and therefore eliminating one variable from the equation.
- Integration by parts is a method that is used in identifying two functions, usually denoted as u and v, and applying the formula ∫u dv = uv - ∫v du.
- The u-substitution formula is ∫f(g(x))g'(x)dx = ∫f(u)du, where u = g(x). This formula helps substitute a new variable u for a function g(x), and then integrate the function f(u) with respect to u.
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Previous Year Questions
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Sample Questions
Ques. Find the integral of the following function f(x), where f(x)= ∫10x (5x2) dx. (3 marks)
Ans. Given f(x) = ∫10x (5x2) dx,
Let F(x) be the integral of the function f(x),
F(x) = ∫f(x) dx = F(x) = ∫10x (5x2) dx
Notice that the function is also a composite function,
Let u = 5x2. Differentiating the expression,
Using these results in the original integral equation,
F(x) = ∫f(x) dx = F(x) = ∫udu
Ques. Find the integral of the following function f(x) where f(x) = 6x (3x2+ 5). (3 marks)
Let F(x) be the integral of the function f(x),
F(x) = ∫f(x)dx = F(x) = ∫F(x) = ∫6x(3x2+ 5) dx
Notice that the function is also a composite function,
Let u = 3x2 + 5. Differentiating the expression,
Using these results in the original integral equation,
F(x) = ∫f(x)dx = F(x) = udu
Ques. Find the integral of the following function f(x), where f(x)= ∫ (3x2+ 6x)(x3+ 3×2+ 5). (3 marks)
Ans. Given f(x) = ∫ (3x2+ 6x)(x3+ 3×2+ 5),
Let F(x) be the integral of the function f(x),
F(x) = ∫f(x) dx = F(x) = ∫ (3x2+ 6x) (x3+ 3×2+ 5) dx
Notice that the function is also a composite function,
Let u = x3 + 3x2 + 5. (Now, Differentiating the expression),
Using these results in the original integral equation,
F(x) = ∫f(x) dx = F(x) = ∫udu
Ques. Find the integral of the following function f(x), where f(x) = sin(x)cos(x). (3 marks)
Ans. Given f(x) = sinx cosx,
Let F(x) be the integral of the function f(x),
F(x) = ∫f(x) dx = F(x) = ∫ sinxcosxdx
Notice that the function is also a composite function,
Let u = cos(x). Differentiating the expression,
Using these results in the original integral equation,
F(x) = ∫f(x) dx = F(x) = ∫ sin(x) cos(x) dx
F(x) = ∫-cos(x)(-sin(x))dx = F(x) = ∫-udu
Ques. Find the integral of the following function f(x), where f(x)= ∫ 3x2cosx3. (4 marks)
Ans. Given, f(x) = ∫ 3x2cosx3,
Let F(x) be the integral of the function f(x),
F(x) = ∫f(x)dx = F(x) = ∫ 3x2cosx3dx
Notice that the function is also a composite function,
Let u = x3. Differentiating the expression,
Using these results in the original integral equation,
F(x) = ∫f(x)dx
F(x) = ∫3x2cos(x)dx
F(x) = ∫cosu du
F(x) = sinu +C
Substituting the value of “u” in the above equation,
F(x) = sin(u) + C
⇒F(x) = sin(x3) + C
Ques. Find the integral of the following function f(x), where f(x) = ∫3x2cosx3. (3 marks)
Ans. Given f(x)= ∫3x2cosx3,
Let F(x) be the integral of the function f(x),
F(x) = ∫f(x)dx
F(x) = ∫3x2cosx3dx
Notice that the function is also a composite function,
Let u = x3. Differentiating the expression,
Using these results in the original integral equation,
F(x) = ∫f(x)dx
F(x)= ∫3x2cos(x)dx
F(x) = ∫cosu du
F(x) = sin(u)+ C
Substituting the value of “u” in the above equation,
F(x) = sin(u) + C
⇒F(x) = sin(x3) + C
Ques. When should you use the Substitution Method of Integration? (2 marks)
Ans. Integration by substitution is also known as the "Reverse Chain Rule" or "U-Substitution Method" in calculus. When an integral value is set up in a unique form, this procedure is used to find it. It denotes that the provided integral is expressed as:
∫ f(k(x)).k'(x).dx = f(u).du
In the example above, we'll first integrate the function in terms of the substituted value (f(u)), and then finish by inserting the original function k. (x).
Ques. What is the method of substitution? (1 mark)
Ans. The method of substitution, also known as u-substitution or change of variables, can simply be defined as a technique used in calculus to calculate integrals.
Ques. Integrate ∫ (2−x) 8dx∫ (2−x) 8dx. (3 marks)
Ans. Let's say u = (2 - x)
As a result, du = (-1)dx.
Substitute the values of u and du in (2x)8dx (2x) 8dx, substituting all forms of x, and you obtain
With the use of the U substitution formula,
8dx = u8 (1) du
= - u 8 d u8du
= - u 9 9 + c u99+c
= - ( 2 x ) 9 9 + c (2x) 99+c - (2 x) 9 9 + c (2x)
= - (2 x ) 9 9 + c (2x)
= - (2 x ) 9 9 + c (2x) 99+c
Ques. Integrate 2x cos (x2 - 5) in terms to x. (3 marks)
Ans. I = ∫∫ 2xcos (x2 - 5)
Let x2 - 5 = t.......(1)
2x. dt = dx
by substituting these numbers
I = ∫∫ cos(t).dt
= sin t + C……………... (2)
Substituting the value of (1) in (2) yields
= sin (x2 - 5) + C.
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