Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter7: Integration
Section7.2: Substitution
Problem 1E: Integration by substitution is related to what differentiation method? What type of integrand...
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Question
![## Evaluating the Integral
Assess the following integral:
\[
\int_{1}^{2} x (x^2 - 1)^8 \, dx
\]
### Explanation
In this problem, we aim to compute the definite integral of the function \( x (x^2 - 1)^8 \) from the lower limit of \(1\) to the upper limit of \(2\).
### Step-by-Step Solution
1. **Substitution Method**: Consider using substitution to simplify the integral. Let:
\[
u = x^2 - 1
\]
Then, differentiate both sides to find \(du\):
\[
\frac{du}{dx} = 2x \implies du = 2x \, dx \implies \frac{du}{2} = x \, dx
\]
2. **Change of Variables**: Substitute \(u\) and \(du\) back into the integral.
- When \( x = 1 \):
\[
u = 1^2 - 1 = 0
\]
- When \( x = 2 \):
\[
u = 2^2 - 1 = 3
\]
Thus, the integral limits change from \(x = 1\) to \(x = 2\) into \(u = 0\) to \(u = 3\), and the integral becomes:
\[
\int_{0}^{3} (u)^8 \cdot \frac{du}{2}
\]
3. **Simplify**: Simplify the integrand before integrating:
\[
\frac{1}{2} \int_{0}^{3} u^8 \, du
\]
4. **Integrate**: Now integrate with respect to \(u\):
\[
\frac{1}{2} \left[ \frac{u^9}{9} \right]_{0}^{3}
\]
5. **Evaluate the Definite Integral**:
\[
\frac{1}{2} \left( \frac{3^9}{9} - \frac{0^9}{9} \right) = \frac{1}{2} \cdot \frac{3^9}{9} = \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe634a2f-90fc-4c70-a49c-ac4a0564f81f%2F3c2372a0-4134-4a73-b0ee-eb32c4cc239c%2Fcm3peyj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Evaluating the Integral
Assess the following integral:
\[
\int_{1}^{2} x (x^2 - 1)^8 \, dx
\]
### Explanation
In this problem, we aim to compute the definite integral of the function \( x (x^2 - 1)^8 \) from the lower limit of \(1\) to the upper limit of \(2\).
### Step-by-Step Solution
1. **Substitution Method**: Consider using substitution to simplify the integral. Let:
\[
u = x^2 - 1
\]
Then, differentiate both sides to find \(du\):
\[
\frac{du}{dx} = 2x \implies du = 2x \, dx \implies \frac{du}{2} = x \, dx
\]
2. **Change of Variables**: Substitute \(u\) and \(du\) back into the integral.
- When \( x = 1 \):
\[
u = 1^2 - 1 = 0
\]
- When \( x = 2 \):
\[
u = 2^2 - 1 = 3
\]
Thus, the integral limits change from \(x = 1\) to \(x = 2\) into \(u = 0\) to \(u = 3\), and the integral becomes:
\[
\int_{0}^{3} (u)^8 \cdot \frac{du}{2}
\]
3. **Simplify**: Simplify the integrand before integrating:
\[
\frac{1}{2} \int_{0}^{3} u^8 \, du
\]
4. **Integrate**: Now integrate with respect to \(u\):
\[
\frac{1}{2} \left[ \frac{u^9}{9} \right]_{0}^{3}
\]
5. **Evaluate the Definite Integral**:
\[
\frac{1}{2} \left( \frac{3^9}{9} - \frac{0^9}{9} \right) = \frac{1}{2} \cdot \frac{3^9}{9} = \
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