Problem 4

Question

Find the indefinite integrels in $1-20$. $\int(x+1)^{-n} d x$

Step-by-Step Solution

Verified

Answer

The integral is $-\frac{(x+1)^{1-n}}{n-1} + C$, assuming $n \neq 1$.

1Step 1: Identify the Integral Formula

For the integral of the form $ \int (x+a)^{-n} dx $, use the formula: $ \int (x+a)^{-n} dx = \frac{(x+a)^{-n+1}}{-n+1} + C $, provided $n eq 1$. In this case, we have $ a = 1 $.

2Step 2: Apply the Formula

Substitute $ n $ with $ n $ and $ a $ with $ 1 $ in the formula: $ \int (x+1)^{-n} dx = \frac{(x+1)^{-n+1}}{-n+1} + C $.

3Step 3: Simplify the Expression

Simplify the expression obtained in Step 2: $ \frac{(x+1)^{-n+1}}{-n+1} + C $ can be further expressed as $-\frac{(x+1)^{1-n}}{n-1} + C$.

4Step 4: Verify Conditions

Check that the assumption $ n eq 1 $ is met to ensure the solution is valid. Since this is the general formula, it applies as long as $ n eq 1 $.

Key Concepts

Power Rule for IntegrationExponential Functions IntegrationIntegration Techniques

Power Rule for Integration

The power rule for integration is a fundamental tool when finding indefinite integrals of polynomials and expressions where the variable is raised to a power. It's similar to the power rule for differentiation, but instead of decreasing the power, we increase it. The integration power rule is expressed as follows for an integral of one term:

$ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $ when $ n eq -1 $

Here, $ n eq -1 $ because, with $ n = -1 $, you find the integral of $ \frac{1}{x} $, which is the natural logarithm function, $ \ln |x| + C $.

In the exercise, we applied a variation to a function that contained $ (x+1) $ as the base instead of simply $ x $. This requires extending the power rule using substitutions. Understanding this extension is crucial for handling more complex expressions involving shifts inside the integrand.

The strategy is to treat $ (x+1)^{-n} $ as a single unit, allowing the power rule to be applied. We adjust the base $ x $ to $ x+a $. This changed our integral and required replacing $ n $ and $ a $ accordingly, which we performed in the solution steps.

Exponential Functions Integration

Integration of exponential functions is often straightforward yet essential. For an integral involving $ e^x $, the function's key property is crucial - it remains unchanged after differentiation or integration. The general formula is:

$ \int e^x \, dx = e^x + C $

Notice this allows exponential functions to integrate naturally and directly, reflecting their continuous growth rate.

Sometimes, more complex expressions with bases other than $ e $ appear, requiring adaptations. Using logarithmic identities or substitutions, we can reduce them into familiar forms. Integrating other base exponential functions $ a^x $ uses:

$ \int a^x \, dx = \frac{a^x}{\ln a} + C $

This expression highlights the role of the natural logarithm in handling non-e exponential bases.

In our original exercise, we focused on power-based integration, but similar principles would apply to exponential functions. Solid understanding of each helps smoothly transition between different function types.

Integration Techniques

When tackling indefinite integrals, a suite of integration techniques becomes invaluable. These methods are like different tools in a toolbox, each designed to handle specific forms and complexities. Here are common techniques beyond the basic power rule:

Substitution: Useful when an integral contains a composite function. Identify a part of the integrand to replace with a single variable, simplifying integration.
Integration by Parts: Breaks down the product of functions into simpler components. Generally employs $ \int u \, dv = uv - \int v \, du $.

For our problem, substitution was crucial as we had $(x+1)$ instead of a simple $x$. Substitution recalibrates the integral to match the form applicable to the integration rule.

Understanding when and how to apply these techniques determines the ease and accuracy of solving integrals. Prerequisites for these include identifying terms fitting the substitution or needing product differentiation. Mastering these methods empowers one to confidently transition across varied integral challenges.

Problem 3

Problem 4

Other exercises in this chapter

Problem 3

Find an antiderivative $f(x)$ for $v(x)$. Then compute the definite integral $\int_{0}^{1} r(x) d x=f(1)-f(0)$. $$ 1 \sqrt{x}\left(\text { or } x^{-1 / 2}

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Problem 3

Evaluate the sum $\sum_{i=0}^{6} 2^{i}$ and $\sum_{i=0}^{n} 2^{i} .$

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Problem 4

About sums $f_{j}$ and differences $\boldsymbol{v}_{j}$. Any constant $C$ can be added to the antiderivative $f(x)$ because the ________ of a constant i

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Problem 5

If $v(x)=d f / d x,$ what constants $C$ make $1-10$ true? $$ \int_{\mathbb{k / 2}}^{b} v(\sin x) \cos x d x=f(\sin b)+C $$

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