Problem 32 Finding an Indefinite Integral I... [FREE SOLUTION]

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Chapter 8: Problem 32

Finding an Indefinite Integral In Exercises $15-34$ , find the indefinite integral. (Note: Solve by the simplest method- not all require integration by parts.) $$\int 4 \arccos x d x$$

Short Answer

Expert verified

The indefinite integral of $4 \arccos x$ is $4x \arccos x + 4\sin^{-1}(x)+ C$.

Step by step solution

Identity selection

Let $u = \arccos x$ and $dv = 4 dx$.

Compute derivatives and integrals

Calculate the derivative $du$ and integral $v$. The derivative of $ \arccos x$ (with respect to $x$) is $-1 / \sqrt{1 - x^2}$, so $du = -1 / \sqrt{1 - x^2} dx$. The integral of $4 dx$ is $4x$, so $v = 4x$.

Apply Integration by parts formula

Apply the formula for integration by parts, $鈭� udv = uv - 鈭� vdu$: \[鈭� 4 \arccos x dx = 4x \arccos x - 鈭� 4x (-1 / \sqrt{1 - x^2}) dx = 4x \arccos x + 鈭� \frac{4x dx}{\sqrt{1 - x^2}}\].

Calculate the integral

We integrate $\frac{4x}{\sqrt{1 - x^2}} dx$ using the power rule of integrals by identifying $a$ in $1-a^2$ inside the square root function. Here $a$ equates to $x$: \[鈭� \frac{4x dx}{\sqrt{1 - x^2}} = 4 鈭� du / \sqrt{1 - u^2} = 4\sin^{-1}(x)+ C\], where $ C $ is the constant of integration.

Final Result

Movement of known expressions back in gives the final result: \[鈭� 4 \arccos x dx = 4x \arccos x + 4\sin^{-1}(x)+ C\].

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Key Concepts

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

Integration by Parts

Integration by parts is a special technique in calculus used to solve integrals that are difficult to handle by standard methods. The formula for integration by parts is \[ \int u \, dv = uv - \int v \, du \], where $ u $ and $ dv $ are parts of the original integral you are trying to solve. The idea is to pick $ u $ and $ dv $ such that differentiating $ u $ (getting $ du $) and integrating $ dv $ (getting $ v $) are straightforward.

**Step 1:** Choose $ u = \arccos x $ and $ dv = 4 \, dx $, since the derivative of $ \arccos x $ is known and integrating a constant is simple.
**Step 2:** Differentiate $ u $ to get $ du = -\frac{1}{\sqrt{1-x^2}} \, dx $, and integrate $ dv $ to find $ v = 4x $.
**Step 3:** Substitute into the integration by parts formula. This means using the expressions $ u $, $ du $, $ v $, and $ dv $ to reformulate the integral: \[ \int 4 \, \arccos x \, dx = 4x \arccos x + \int \frac{4x}{\sqrt{1-x^2}} \, dx \].

This transformation simplifies solving the original problem by breaking it down into simpler parts.

Arccosine Function

The arccosine function is the inverse of the cosine function. It is written as $ \arccos(x) $, which means it returns the angle whose cosine is $ x $. This specific function is widely used when dealing with trigonometric equations and integrals.

Key Points about Arccosine:

**Definition:** $ \arccos(x) $ yields an angle $ \theta $ such that $ \cos(\theta) = x $, where $ 0 \leq \theta \leq \pi $.
**Derivative:** The derivative of $ \arccos(x) $ is given by $ -\frac{1}{\sqrt{1-x^2}} $. This negative derivative reflects the slope of its curve.
**Range:** The output of $ \arccos(x) $ is constrained between $ 0 $ and $ \pi $, fitting the range of possible x-values for which cosine can give non-negative results.

In integrals, the appearance of $ \arccos(x) $ often necessitates methods like integration by parts, which help seamlessly integrate such trigonometric functions.

Integral Calculus

Integral calculus is a fundamental branch of calculus focusing on the concept of integration, primarily used to calculate areas under curves. Indefinite integrals, like the one in our exercise, represent a family of functions and include a constant of integration, $ C $.

Key Concepts of Integral Calculus:

**Purpose:** Integral calculus is essentially the reverse process of differentiation. It is concerned with finding functions whose derivative is a specified function, known as antiderivatives.
**Notation:** An indefinite integral takes the form $ \int f(x) \, dx $, with the integral sign indicating the process applied to the function $ f(x) $.
**Equation:** In solving $ \int 4 \arccos x \, dx $, we utilized techniques like integration by parts, illustrating how integration isn't always straightforward and often demands strategic methods to break complex expressions into manageable parts.

With a firm understanding of integral calculus, you can tackle various mathematical and real-world problems by finding areas, volumes, and other quantities that accumulate continuously.

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91影视

Finding an Indefinite Integral In Exercises \(15-34\) , find the indefinite integral. (Note: Solve by the simplest method- not all require integration by parts.) $$\int 4 \arccos x d x$$

Short Answer

Step by step solution

Identity selection

Compute derivatives and integrals

Apply Integration by parts formula

Calculate the integral

Final Result

Key Concepts

Integration by Parts

Arccosine Function

Integral Calculus

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