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Magazine Chapter 9: Problem 3
Evaluate the following integrals: $$\int x(x+7)^{4} d x$$
Short Answer
Expert verified
\[ \frac{(x+7)^6}{6} - \frac{7(x+7)^5}{5} + C \]
Step by step solution
01
- Use the substitution method
Define a substitution for the integral. Let: \[ u = x+7 \] Then, derive the differential of \( u \): \[ du = dx \] Also, rewrite \( x \) in terms of \( u \): \[ x = u - 7 \].
02
- Substitute the variables
Replace \( x \) and \( dx \) in the integral with \( u \) and \( du \): \[ \int x(x+7)^{4} \, dx = \int (u-7) u^{4} \, du \] Simplify the integral: \[ \int (u-7) u^{4} \, du = \int (u^5 - 7u^4) \, du \].
03
- Integrate term by term
Evaluate the integral by integrating each term separately: \[ \int u^5 \, du - \int 7u^4 \, du \]. The integrals are: \[ \int u^5 \, du = \frac{u^6}{6} \] \[ \int 7u^4 \, du = 7 \int u^4 \, du = 7 \cdot \frac{u^5}{5} = \frac{7u^5}{5} \].
04
- Combine the results and revert the substitution
Combine and simplify the results of the individual integrals: \[ \frac{u^6}{6} - \frac{7u^5}{5} + C \] Finally, substitute back \( u = x + 7 \) to express the integral in terms of \( x \): \[ \frac{(x+7)^6}{6} - \frac{7(x+7)^5}{5} + C \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method, also known as \(u\)-substitution, is a crucial technique in integral calculus. It simplifies evaluating complex integrals by changing variables.
Let's break it down:
Let's break it down:
- First, choose a substitution that simplifies the integral. For example, if the integrand is complex, look for expressions that repeat themselves.
- In our example, we let \(u = x + 7\). This simplifies the integrand because the expression \((x+7)\) appears multiple times.
- Next, derive the differential of \(u\): \(du = dx\). This helps in replacing \(dx\) in the integrand.
- Rewriting the integral in terms of \(u\) results in an easier integral to solve.
Evaluating Integrals
Evaluating integrals is a fundamental skill in calculus, allowing us to find areas under curves and solve various physical problems.
Here are some key points:
Here are some key points:
- After substituting the variables in our example, we obtained \(\forall \text{ }\frac{(u^5 - 7u^4) \text{ } du}\).
- The next step is to break this integral into simpler parts.
- Use formulas like \(\forall \text{ }\frac{\frac{{u^n}}{{n+1}}}+ \text{ constant}\) for powers of \(u\).
Term-by-Term Integration
Term-by-term integration involves breaking down a complex integral into smaller, manageable parts. This technique simplifies solving integrals significantly.
For our example:
Term-by-term integration helps to simplify and solve integrals faster. Always look for opportunities to decompose the integral to make your calculations easier.
For our example:
- First, decompose \(\forall (u^5 - 7u^4) \text{ } du\) into \(\forall \text{ }\frac{{u^5 \text{ } du}} - 7 \forall \text{ } \frac{u^4 \text{ } du}\).
- For \(\forall \text{ }\frac{u^5 \text{ } du}\), we get \(\frac{u^6}{6}\).
- For \(\forall \text{ }7 \frac{u^4 \text{ }du}\), we get \(\frac{7u^5}{5}\).
Term-by-term integration helps to simplify and solve integrals faster. Always look for opportunities to decompose the integral to make your calculations easier.
Definite Integral
Unlike indefinite integrals that include a constant of integration (\(C\)), definite integrals calculate the exact area under a curve between specified limits.
Perform these steps:
Definite integrals are essential in real-life applications like physics, engineering, and statistics, where calculations of exact areas or accumulated quantities are required.
Perform these steps:
- Evaluate the integral normally.
- After integrating, substitute the upper and lower bounds into the integrated function.
- Finally, subtract the value obtained at the lower limit from the value at the upper limit.
Definite integrals are essential in real-life applications like physics, engineering, and statistics, where calculations of exact areas or accumulated quantities are required.
