Textbook Question
68. Different methods
b. Evaluate ∫(cot x csc² x) dx using the substitution u=cscx.
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7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
3:21 minutesProblem 4.R.93
Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ ((1/x²) - (2/(x⁵⸍²))) dx
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Rewrite the integrand to simplify the terms. The given integral is ∫ ((1/x²) - (2/(x^(5/2)))) dx. Express each term using exponents: ∫ (x^(-2) - 2x^(-5/2)) dx.
Apply the power rule for integration. Recall that the integral of x^n is (x^(n+1))/(n+1), provided n ≠ -1. Use this rule for each term in the integrand.
For the first term, x^(-2), integrate it using the power rule: ∫ x^(-2) dx = (x^(-2+1))/(-2+1) = x^(-1)/(-1) = -1/x.
For the second term, -2x^(-5/2), integrate it using the power rule: ∫ -2x^(-5/2) dx = -2 * (x^(-5/2+1))/(-5/2+1) = -2 * (x^(-3/2))/(-3/2). Simplify the coefficient.
Combine the results of the two integrals and add the constant of integration, C, to express the final indefinite integral: ∫ ((1/x²) - (2/(x^(5/2)))) dx = -1/x + (4/3)x^(-3/2) + C.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integrals
Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits of integration and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation, where we seek a function F(x) such that F'(x) equals the integrand.
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Power Rule for Integration
The power rule for integration is a fundamental technique used to integrate functions of the form x^n, where n is any real number except -1. According to this rule, the integral of x^n is (x^(n+1))/(n+1) + C. This rule simplifies the process of integrating polynomial and rational functions, making it essential for solving many integral problems.
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Rational Functions
Rational functions are ratios of two polynomials, expressed as P(x)/Q(x), where P and Q are polynomials. When integrating rational functions, techniques such as polynomial long division or substitution may be necessary to simplify the integrand. Understanding how to manipulate and integrate these functions is crucial for solving integrals involving terms like 1/x² and 2/x⁵.
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