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Chapter 8: Problem 77

In each case, check your work by differentiating your answer with respect to \(x\). $$\int \sinh ^{-1} x d x$$.

Short Answer

Expert verified
The integral is \( x \sinh^{-1} x - \sqrt{x^2+1} + C \).

Step by step solution

01

Identify Integration Formula

To integrate \(\sinh^{-1}x \), we recognize that it follows the standard integration formula for inverse hyperbolic sine functions: \[ \int \sinh^{-1} x \, dx = x \sinh^{-1} x - \sqrt{x^2+1} + C \] where \(C\) is the constant of integration.
02

Apply the Integration Formula

Using the formula from Step 1, the integral of \(\sinh^{-1} x\) with respect to \(x\) becomes: \[ \int \sinh^{-1} x \, dx = x \sinh^{-1} x - \sqrt{x^2+1} + C \] where \(C\) is the constant of integration.
03

Differentiate the Result to Verify

Differentiate the expression \(x \sinh^{-1} x - \sqrt{x^2+1} + C\) with respect to \(x\). Apply the product rule to \(x \sinh^{-1} x\) and the chain rule to \(\sqrt{x^2+1}\). The derivative of \(x \sinh^{-1} x\) is \(\sinh^{-1} x + x \cdot \frac{1}{\sqrt{x^2+1}}\), and the derivative of \(-\sqrt{x^2+1}\) is \(-\frac{x}{\sqrt{x^2+1}}\).So, the derivative of the entire expression is: \[ \sinh^{-1} x + x \cdot \frac{1}{\sqrt{x^2+1}} - \frac{x}{\sqrt{x^2+1}} = \sinh^{-1} x \]This confirms that the original integral is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration Techniques
Integration techniques are methods used to find the integral of a function. In this exercise, the focus is on integrating the inverse hyperbolic sine function, \( \sinh^{-1}x \). The integration of inverse hyperbolic functions often involves recognizing patterns and applying known formulas.
  • Recognize the Function: First, identify the function you need to integrate. Here it's the inverse hyperbolic sine, \( \sinh^{-1}x \).
  • Use a Standard Formula: The specific formula used here is \( \int \sinh^{-1}x\, dx = x \sinh^{-1}x - \sqrt{x^2+1} + C \).
This formula helps us to bypass complex calculations and directly integrate the function.
This approach is a shortcut that requires memorizing common integral forms, especially useful for trigonometric and hyperbolic functions.
Differentiation
Differentiation is the process of finding the derivative of a function. It shows how fast a function is changing at any point. To check our integration work, we differentiate the result. Here, the function to differentiate is:
  • \(x \sinh^{-1} x - \sqrt{x^2+1} + C\)
Differentiating requires familiarity with several rules:
  • Basic Derivative Rules: Knowing how to differentiate power functions, constants, and basic operations.
  • Chain Rule: Used when differentiating composite functions like \( \sqrt{x^2+1} \).
Overall, the final differentiation verifies that we recover the original integrand, \( \sinh^{-1}x \), confirming the correctness of our integration.
Product Rule
The product rule is a fundamental rule in differentiation allowing us to find the derivative of products of two functions. If \( u(x) \) and \( v(x) \) are two functions of \( x \), the product rule states:\[(uv)' = u'v + uv'\]In this integration check, apply the product rule to differentiate the term \( x \sinh^{-1} x \). Consider:
  • \( u = x \) and \( v = \sinh^{-1}x \)
This gives us:
  • \( u' = 1 \)
  • \( v' = \frac{1}{\sqrt{x^2+1}} \)
Applying the product rule, we get:\[(x \sinh^{-1} x)' = 1 \cdot \sinh^{-1} x + x \cdot \frac{1}{\sqrt{x^2+1}}\]Thus, applying this rule correctly confirms the differentiation part needed for verification of the integral results.

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Most popular questions from this chapter

Evaluate the integrals in Exercises \(39-50\). $$\begin{array}{l} \int \frac{1}{\left(x^{1 / 3}-1\right) \sqrt{x}} d x \\ \text { (Hint: Let } \, x=u^{6} \text { .) } \end{array}$$

Expand the quotients in Exercises \(1-8\) by partial fractions. $$\frac{z}{z^{3}-z^{2}-6 z}$$

a. Show that $$\int_{3}^{\infty} e^{-3 x} d x=\frac{1}{3} e^{-9}<0.000042$$ and hence that \(\int_{3}^{\infty} e^{-x^{2}} d x<0.000042 .\) Explain why this means that \(\int_{0}^{\infty} e^{-x^{2}} d x\) can be replaced by \(\int_{0}^{3} e^{-x^{2}} d x\) without introducing an error of magnitude greater than 0.000042 b. Evaluate \(\int_{0}^{3} e^{-x^{2}} d x\) numerically.

Evaluate the integrals in Exercises \(39-50\). $$\int \frac{(x-2)^{2} \tan ^{-1}(2 x)-12 x^{3}-3 x}{\left(4 x^{2}+1\right)(x-2)^{2}} d x$$

In Exercises \(33-38,\) perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral. $$\int \frac{9 x^{3}-3 x+1}{x^{3}-x^{2}} d x$$
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