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Indefinite integral of 1/cos^6x sin^6x?
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Indefinite integral of 1/cos^6x sin^6x?
Indefinite Integral of 1/cos^6x sin^6x
When finding the indefinite integral of a function, we aim to find the antiderivative of the function. In other words, we look for a function whose derivative is the given function. In this case, we are given the function 1/cos^6x sin^6x and we need to find its indefinite integral.
Step 1: Simplify the Function
Before we can integrate the function, we need to simplify it. We can do this by using trigonometric identities. We know that:
- cos^2x + sin^2x = 1
- 1/cos^2x = sec^2x
- 1/sin^2x = cosec^2x
Using these identities, we can simplify 1/cos^6x sin^6x as follows:
1/cos^6x sin^6x = (1/cos^2x)^3 (1/sin^2x)^3
Substituting the identities for 1/cos^2x and 1/sin^2x:
(sec^2x)^3 (cosec^2x)^3 = sec^6x cosec^6x
So, 1/cos^6x sin^6x = sec^6x cosec^6x
Step 2: Use the Power Reduction Formula
Now that we have simplified the function, we can use the power reduction formula to integrate it. The power reduction formula is:
sin^m x cos^n x dx = (1/2) sin^(m-1) x cos^(n+1) x dx - (m-1)/(2n+2) sin^(m-3) x cos^(n+2) x dx
Using this formula with m = n = 3, we get:
sec^6x cosec^6x dx = (1/2) sec^3x cosec^7x dx - (3/10) sec^3x cosec^9x dx
Step 3: Integrate the Function
Now that we have simplified the function and applied the power reduction formula, we can integrate it. Integrating the first term:
(1/2) sec^3x cosec^7x dx = (1/2) ∫ sec^3x cosec^7x dx
We can use the substitution u = sin x to solve this integral. Using the identity 1 + tan^2x = sec^2x, we can write:
∫ sec^3x dx = ∫ sec^3x (1 + tan^2x)/ (1 + tan^2x) dx
Substituting u = sin x:
∫ sec^3x dx = ∫ (1 + tan^2x)/ (
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Indefinite integral of 1/cos^6x sin^6x?
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Indefinite integral of 1/cos^6x sin^6x? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Indefinite integral of 1/cos^6x sin^6x? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Indefinite integral of 1/cos^6x sin^6x?.
Indefinite integral of 1/cos^6x sin^6x? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Indefinite integral of 1/cos^6x sin^6x? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Indefinite integral of 1/cos^6x sin^6x?.
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