Textbook Question
71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.
74. ∫xⁿ arcsin(x) dx (Hint: integration by parts.)
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12. Techniques of Integration
Integration by Parts
3:16 minutesProblem 8.R.60
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
60. ∫ x² coshx dx
Verified step by step guidance1
Step 1: Recognize that the integral ∫ x² cosh(x) dx involves a product of a polynomial (x²) and a hyperbolic function (cosh(x)). This suggests that the method of integration by parts is appropriate. Recall the formula for integration by parts: ∫ u dv = uv - ∫ v du.
Step 2: Choose u and dv wisely. Let u = x² (the polynomial term, which simplifies upon differentiation) and dv = cosh(x) dx (the hyperbolic function, which is straightforward to integrate). Compute du = 2x dx and v = sinh(x), since the integral of cosh(x) is sinh(x).
Step 3: Apply the integration by parts formula: ∫ x² cosh(x) dx = u * v - ∫ v * du. Substitute u = x², v = sinh(x), and du = 2x dx into the formula to get: x² sinh(x) - ∫ 2x sinh(x) dx.
Step 4: Notice that the remaining integral ∫ 2x sinh(x) dx still involves a product of a polynomial and a hyperbolic function. Apply integration by parts again. Let u = 2x and dv = sinh(x) dx. Compute du = 2 dx and v = -cosh(x), since the integral of sinh(x) is -cosh(x).
Step 5: Substitute into the integration by parts formula for the second integral: ∫ 2x sinh(x) dx = u * v - ∫ v * du. This becomes 2x(-cosh(x)) - ∫ -cosh(x) * 2 dx. Simplify the expression and combine terms to complete the solution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration Techniques
Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric identities. Understanding these methods is crucial for solving integrals that cannot be evaluated using basic antiderivatives.
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Hyperbolic Functions
Hyperbolic functions, such as cosh(x), are analogs of trigonometric functions but are based on hyperbolas rather than circles. They are defined using exponential functions, with cosh(x) = (e^x + e^(-x))/2. Recognizing these functions and their properties is essential for evaluating integrals involving them.
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Integration by Parts
Integration by parts is a technique derived from the product rule of differentiation. It is used to integrate products of functions and is expressed as ∫u dv = uv - ∫v du. This method is particularly useful when dealing with integrals that involve polynomial and hyperbolic functions, as in the given problem.
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