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Magazine Chapter 5: Problem 71
Evaluate. $$ \int_{0}^{\sqrt{7}} 7 x \sqrt[3]{1+x^{2}} d x $$
Short Answer
Expert verified
39.375
Step by step solution
01
Apply Substitution
First, let us consider a substitution to simplify the integral. Set \( u = 1 + x^2 \). Then, compute the differential: \( du = 2x \, dx \), or equivalently \( x \, dx = \frac{1}{2} \, du \).
02
Change the Limits of Integration
The next step is to change the limits of integration based on the new variable \( u \). When \( x = 0 \), \( u = 1 \). When \( x = \sqrt{7} \), \( u = 1 + (\sqrt{7})^{2} = 8 \).
03
Rewrite the Integral
Substitute the new variable and differential into the integral:\[\int_{0}^{\sqrt{7}} 7 x \sqrt[3]{1+x^{2}} \, dx = 7 \int_{1}^{8} \sqrt[3]{u} \frac{1}{2} \, du \]
04
Simplify the Integral
Simplify the constants and the integral:\[\int_{1}^{8} 7 * \frac{1}{2} \sqrt[3]{u} \, du = \frac{7}{2} \int_{1}^{8} u^{1/3} \, du \]
05
Evaluate the Integral
Now, integrate \( u^{1/3} \):\[\int u^{1/3} \, du = \frac{u^{4/3}}{4/3} = \frac{3}{4} u^{4/3} \]Thus, the integral becomes:\[\frac{7}{2} * \frac{3}{4} * \[ u^{4/3} \]_{1}^{8} = \frac{21}{8} [ u^{4/3} ]_{1}^{8} \]
06
Compute the Definite Integral
Plug in the limits of integration:\[\frac{21}{8} [ 8^{4/3} - 1^{4/3} ] = \frac{21}{8} [ 16 - 1] = \frac{21}{8} * 15 = 39.375 \]
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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 is a key technique used in calculus to simplify integrals. It involves substituting a part of the integrand with a new variable in order to make the integral easier to solve. In our exercise, we set:
- \( u = 1 + x^2 \)
- Compute the differential: \( du = 2x \, dx \)
- Rewriting \( x \, dx \) gives \( x \, dx = \frac{1}{2} \, du \)
changing limits of integration
When performing substitutions, it's important to change the limits of integration to match the new variable. In the provided exercise:
- Original limits are from \( x = 0 \) to \( x = \sqrt{7} \)
- By substituting: \( u = 1 + x^2 \)
- When \( x = 0 \), \( u = 1 \)
- When \( x = \sqrt{7} \), \( u = 1 + 7 = 8 \)
evaluating definite integrals
Evaluating definite integrals involves solving the integral within the given limits. After substituting and changing limits:
We had:
\[ 7 \int_{1}^{8} \sqrt[3]{u} \frac{1}{2} \, du \]
Simplify to obtain:
\[ \frac{7}{2} \int_{1}^{8} u^{1/3} \, du \]
Integrate \( u^{1/3} \) to get:
\[ \frac{u^{4/3}}{4/3} = \frac{3}{4} u^{4/3} \]
Plug in the limits of integration:
\[ \frac{21}{8} \left[ u^{4/3} \right]_{1}^{8} = \frac{21}{8} \left[ 8^{4/3} - 1^{4/3} \right] \]
which simplifies to:
\[ \frac{21}{8} \left[ 16 - 1 \right] = \frac{21}{8} * 15 = 39.375 \]
We had:
\[ 7 \int_{1}^{8} \sqrt[3]{u} \frac{1}{2} \, du \]
Simplify to obtain:
\[ \frac{7}{2} \int_{1}^{8} u^{1/3} \, du \]
Integrate \( u^{1/3} \) to get:
\[ \frac{u^{4/3}}{4/3} = \frac{3}{4} u^{4/3} \]
Plug in the limits of integration:
\[ \frac{21}{8} \left[ u^{4/3} \right]_{1}^{8} = \frac{21}{8} \left[ 8^{4/3} - 1^{4/3} \right] \]
which simplifies to:
\[ \frac{21}{8} \left[ 16 - 1 \right] = \frac{21}{8} * 15 = 39.375 \]
simplifying integrals
Simplifying integrals helps in making additions or multiplications inside the integral more manageable. In the example:
We rewrote the integral after substitution as:
\[ \frac{7}{2} \int_{1}^{8} u^{1/3} \, du \]
This allowed us to focus on integrating \( u^{1/3} \). Then we integrated:
\[ \frac{u^{4/3}}{4/3} \]
Finally, we used multipliers and constants outside the integral to simplify the expression further. Simplification often involves:
We rewrote the integral after substitution as:
\[ \frac{7}{2} \int_{1}^{8} u^{1/3} \, du \]
This allowed us to focus on integrating \( u^{1/3} \). Then we integrated:
\[ \frac{u^{4/3}}{4/3} \]
Finally, we used multipliers and constants outside the integral to simplify the expression further. Simplification often involves:
- Breaking integrals into simpler parts
- Reducing constants
- Integrating standard formulas
- Applying limits properly
