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Chapter 5: Problem 66

Evaluate. Each of the following can be integrated using the rules developed in this section, but some algebra may be required beforehand. $$ \int(5 t+4)^{2} d t $$

Short Answer

Expert verified
\( \frac{25t^3}{3} + 20t^2 + 16t + C \)

Step by step solution

01

- Expand the integrand

First, expand \( (5t + 4)^2 \) using the binomial theorem: \[ (5t + 4)^2 = 25t^2 + 40t + 16 \]
02

- Write the expanded form inside the integral

Now, integrate the expanded form: \[ \int (25t^2 + 40t + 16) \, dt \]
03

- Integrate term-by-term

Integrate each term separately: \[ \int 25t^2 \, dt + \int 40t \, dt + \int 16 \, dt \]
04

- Apply the power rule

Use the power rule \( \int t^n \, dt = \frac{t^{n+1}}{n+1} \): \[ 25 \int t^2 \, dt = 25 \cdot \frac{t^3}{3} = \frac{25t^3}{3} \] \[ 40 \int t \, dt = 40 \cdot \frac{t^2}{2} = 20t^2 \] \[ 16 \int 1 \, dt = 16t \]
05

- Write the final integrated result

Combine the integrated results and add the constant of integration \( C \): \[ \frac{25t^3}{3} + 20t^2 + 16t + C \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule in Integration
The power rule is a fundamental concept in integral calculus. It's often the first technique students learn when performing basic integrations. The power rule states:

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