Integration is a fundamental concept in calculus that allows us to find the antiderivative of a function. One powerful technique for solving integrals is called integration by substitution, commonly known as ILATE. Integration by substitution is also termed as the full form of ILATE. This method allows us to simplify complicated integrals by substituting variables. By choosing appropriate substitutions and following the rule and formula, we can transform challenging integrals into more manageable forms. Through practice and understanding, mastering ILATE enables us to solve a wide range of integration problems efficiently.
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The ILATE Rule (Integration By Parts Rule):
The ILATE Rule In integration, ILATE, can be stated as follows:
∫f(g(x)) * g'(x) dx = ∫f(u) du (ILATE Rule Formula)
(where u = g(x) and du = g'(x) dx. In other words, we substitute the variable u for g(x) and differentiate to find du. Then we rewrite the integral in terms of u and du)
In other words, this rule states that if we have a function f(g(x)) multiplied by the derivative of the inner function g(x), denoted as g'(x), we can rewrite the integral in terms of the new variable u = g(x) and its differential du = g'(x) dx. By doing so, we simplify the integral and make it easier to evaluate.
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Derivation of Integration By Parts Formula
The integration by-parts formula can be derived using the following steps:
- Let u(x) and v(x) be any two functions.
- Let ∫ u(x) v'(x) dx be the integral we want to evaluate.
- Let w(x) = ∫ u(x) v'(x) dx.
- Differentiate both sides of the equation w(x) = ∫ u(x) v'(x) dx to get w'(x) = u(x) v'(x).
- Integrate both sides of the equation w'(x) = u(x) v'(x) to get w(x) = u(x) v(x) - ∫ u'(x) v(x) dx.
- Substitute w(x) back into the equation w(x) = u(x) v(x) - ∫ u'(x) v(x) dx to get the integration by parts formula:
- ∫ u(x) v'(x) dx = u(x) v(x) - ∫ u'(x) v(x) dx
The integration by parts formula can be used to evaluate integrals of the product of two functions. The first function, u(x), is called the integrand, and the second function, v'(x), is called the integrand's derivative. The integral of the product of two functions is easier to evaluate if the first function is easier to differentiate.
The ILATE Formula (Integration By Parts Formula):
To apply ILATE formula for integration, follow these steps:
1. Identify a function within the integral that can be written as the composition of two functions, such as f(g(x)).
2. Let u be the inner function, g(x), and differentiate it to find du.
3. Rewrite the integral in terms of u and du.
4. Evaluate the new integral in terms of u.
5. If necessary, convert the result back to the original variable x.
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ILATE Examples (Illustrating ILATE Integration)
Let's look at a few examples to illustrate the application of ILATE/ILATE integration formula:
Example 1:
Evaluate ∫x^2 * (x^3 + 1)^4 dx.
In this example, we can let u = x^3 + 1. Differentiating, we get du = 3x^2 dx. Rearranging, dx = du / (3x^2).
Substituting u and dx in terms of u, the integral becomes:
∫(u - 1)^4 * (du / (3x^2))
Simplifying further:
(1/3)∫(u - 1)^4 / x^2 du
Now, integrate with respect to u:
(1/3) * (u^5/5 - 4u^4/4 + 6u^3/3 - 4u^2/2 + u) + C
Finally, substitute u back in terms of x to get the final answer.
Example 2:
Evaluate ∫x * e^(x^2) dx.
In this example, we can let u = x^2. Differentiating, we get du = 2x dx. Rearranging, dx = du / (2x).
Substituting u and dx in terms of u, the integral becomes:
(1/2)∫e^u du
Now, integrate with respect to u:
(1/2) * e^u + C
Finally, substitute u back in terms of x to obtain the final answer.
Example 3:
Evaluate ∫x^2 * e^(x^3) dx.
In this example, we can let u = x^3. Differentiating, we get du = 3x^2 dx. Rearranging, dx = du / (3x^2).
Substituting u and dx in terms of u, the integral becomes:
(1/3) ∫e^u du
Now, integrate with respect to u:
(1/3) * e^u + C
Finally, substitute u back in terms of x to obtain the final answer.
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Example 4:
Evaluate ∫x * sqrt(x^2 + 1) dx.
In this example, we can let u = x^2 + 1. To find du, we differentiate u with respect to x: du = 2x dx. Rearranging, we have dx = du / (2x).
Substituting u and dx in terms of u, the integral becomes:
(1/2) ∫sqrt(u) du
Now, integrate with respect to u:
(1/2) * (2/3) * u^(3/2) + C
Finally, substitute u back in terms of x to obtain the final answer.
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Example 5:
Evaluate ∫x * sin(x^2) dx.
In this example, we can let u = x^2. Differentiating, we get du = 2x dx. Rearranging, dx = du / (2x).
Substituting u and dx in terms of u, the integral becomes:
(1/2) ∫sin(u) du
Now, integrate with respect to u:
-(1/2) * cos(u) + C
Finally, substitute u back in terms of x to obtain the final answer.