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Magazine Chapter 6: Problem 48
Find in two different ways and check that your answers agree. \(\int x(x+4)^{6} d x\) a. Use integration by parts. b. Use the substitution \(u=x+4\) (so \(x\) is replaced by \(u-4\) ) and then multiply out the integrand.
Short Answer
Expert verified
Both methods yield the integral \( \frac{(x+4)^8}{8} - \frac{4(x+4)^7}{7} + C \).
Step by step solution
01
Integration by Parts Formula
Integration by parts is given by the formula \( \int u \, dv = uv - \int v \, du \). In this context, let's choose \( u = x \) and \( dv = (x+4)^6 \, dx \).
02
Differentiate and Integrate Components for Integration by Parts
Differentiate \( u \) to get \( du = dx \). Integrate \( dv \) to get \( v = \int (x+4)^6 \, dx = \frac{(x+4)^7}{7} \).
03
Apply Integration by Parts Formula
Apply the integration by parts formula: \[ \int x(x+4)^6 \, dx = x \cdot \frac{(x+4)^7}{7} - \int \frac{(x+4)^7}{7} \, dx \].
04
Simplify the Integral After Integration by Parts
The second integral needs to be computed: \[ \int \frac{(x+4)^7}{7} \, dx = \frac{1}{7} \cdot \frac{(x+4)^8}{8} + C = \frac{(x+4)^8}{56} + C \]. Thus, the solution by integration by parts is \[ \frac{x(x+4)^7}{7} - \frac{(x+4)^8}{56} + C \].
05
Substitution Method Setup
With substitution, let \( u = x+4 \), thus \( du = dx \) and \( x = u - 4 \). The integral becomes \( \int (u-4)u^6 \, du \).
06
Expand the Integrand
Expand \( (u-4)u^6 = u^7 - 4u^6 \). Thus, your integral is \( \int (u^7 - 4u^6) \, du \).
07
Integrate the Polynomial
Integrate both terms separately: \( \int u^7 \, du = \frac{u^8}{8} \) and \( \int -4u^6 \, du = -\frac{4u^7}{7} \). Combine these to get \( \frac{u^8}{8} - \frac{4u^7}{7} + C \).
08
Substitute Back for x
Replace \( u \) with \( x+4 \) in the result, yielding \( \frac{(x+4)^8}{8} - \frac{4(x+4)^7}{7} + C \). This should match the result obtained by integration by parts upon simplification.
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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 handy technique used to solve integrals where direct integration is difficult. It works using the formula: \( \int u \, dv = uv - \int v \, du \).
Here, we strategically choose parts of the integrand to simplify the process.
Here, we strategically choose parts of the integrand to simplify the process.
- Choosing \( u \): Select a function that becomes simpler when differentiated. For example, in our exercise, \( u = x \) simplifies to \( du = dx \).
- Choosing \( dv \): Select a function that is straightforward to integrate. Here, \( dv = (x+4)^6 \, dx \) becomes \( v = \frac{(x+4)^7}{7} \) when integrated.
Substitution Method
The substitution method, also known as "u-substitution," is similar to the chain rule in reverse. It simplifies an integral by transforming the variable, making it easier to solve.
To apply it, you follow these steps:
To apply it, you follow these steps:
- Select a substitution: Define \( u \) as a function of \( x \) to replace a difficult part of the integrand. In the example, setting \( u = x + 4 \) simplifies the expression significantly.
- Find \( du \): Differentiate the substitution with respect to \( x \), giving \( du = dx \). This helps in converting the entire integral in terms of \( u \).
- Transform the integral: Express \( x \) in terms of \( u \) and re-write the entire integral in terms of \( u \). You will now have new limits of integration if it's a definite integral, not in our case, making it simpler to solve.
Definite Integrals
A definite integral, unlike an indefinite integral, computes the net area under a curve between two points. This results in a specific numerical value rather than a general function plus a constant \( C \).
The typical notation of a definite integral is \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration. Here's how to handle them:
The typical notation of a definite integral is \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration. Here's how to handle them:
- Integration: First, solve the integral as you would for an indefinite integral, ignoring limits. You find the antiderivative \( F(x) \).
- Apply Limits: Use the Fundamental Theorem of Calculus that says \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \). Subtract the antiderivative function at two boundary points.
- Compute: Calculate the difference \( F(b) - F(a) \) to get the final result. This computation represents the total accumulation of quantities, such as area, between the two limits.
