Problem 69 Evaluate. Each of the following ... [FREE SOLUTION]

91影视

Calculus for Life Sciences

Marvin L. Bittinger, Neal Brand, John Quintanill

Biology

1 Edition

Chapter 5: Problem 69

Evaluate. Each of the following can be integrated using the rules developed in this section, but some algebra may be required beforehand. $$ \int \frac{(t+3)^{2}}{\sqrt{t}} d t $$

Short Answer

Expert verified

$ \frac{2}{5}t^{5/2} + 4t^{3/2} + 18t^{1/2} + C $

Step by step solution

Expand the numerator

First, expand $ (t+3)^2 $ using the binomial theorem. \[ (t+3)^2 = t^2 + 6t + 9 \]

Rewrite the integral

Rewrite the integral by substituting the expanded form of $ (t+3)^2 $: \[ \int \frac{t^2 + 6t + 9}{\sqrt{t}} \, dt \]

Simplify the integrand

Simplify each term in the numerator by dividing by $ \sqrt{t} $: \[ \int \left( \frac{t^2}{\sqrt{t}} + \frac{6t}{\sqrt{t}} + \frac{9}{\sqrt{t}} \right) \, dt = \int \left( t^{3/2} + 6t^{1/2} + 9t^{-1/2} \right) \, dt \]

Integrate each term separately

Integrate each term separately using the power rule: \[ \int t^{3/2} \, dt + 6 \int t^{1/2} \, dt + 9 \int t^{-1/2} \, dt \]

Apply the power rule

For each term, apply the power rule for integration $ \int t^n \, dt = \frac{t^{n+1}}{n+1} + C $: \[ \int t^{3/2} \, dt = \frac{2}{5}t^{5/2}, \ \int t^{1/2} \, dt = \frac{2}{3}t^{3/2}, \ \int t^{-1/2} \, dt = 2t^{1/2} \]

Substitute back into the integral

Combine the integrals with the constants calculated in the previous step: \[ \int (t^{3/2} + 6t^{1/2} + 9t^{-1/2}) \, dt = \frac{2}{5}t^{5/2} + 6 \cdot \frac{2}{3}t^{3/2} + 9 \cdot 2t^{1/2} = \frac{2}{5}t^{5/2} + 4t^{3/2} + 18t^{1/2} + C \]

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Key Concepts

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

Definite Integral

Understanding the definite integral is crucial in calculus. A definite integral represents the signed area under a curve between two specific points. It's written as $\int_{a}^{b} f(x) \, dx$. The result is a number that indicates the net area between $f(x)$ and the x-axis from $x = a$ to $x = b$. Key steps to solve a definite integral include:

Understanding the function to be integrated.
Determining the limits $a$ and $b$.
Calculating the antiderivative $F(x)$.
Evaluating $F(b) - F(a)$.

Take your time practicing with different functions. This will help solidify the concept of definite integrals.

Indefinite Integral

Indefinite integrals are another vital part of calculus. They represent a family of functions and are written without specific limits. The general form is $\int f(x) \, dx$. Instead of a numerical result, you get a function plus an arbitrary constant $C$, since integration reverses differentiation and loses specific values. Let's break down the process:

Identify the function to integrate.
Find the antiderivative.
Add the constant of integration $C$.

For example, integrating $f(x) = x^2$ gives $F(x) = \frac{x^{3}}{3} + C$. Practice finding indefinite integrals with various functions. This will strengthen your skills and understanding.

Binomial Theorem

The binomial theorem is a powerful tool for expanding expressions raised to a power. It鈥檚 especially useful in algebra and calculus. The binomial theorem states:

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \]

$a$ and $b$ are any numbers or variables.
$n$ is a non-negative integer.
$\binom{n}{k}$ is a binomial coefficient showing combinations.

For example, to expand $ (t+3)^2 $ using the binomial theorem:

$a = t $, $b = 3 $, $n = 2$.
$\binom{2}{0} t^2 + \binom{2}{1} t^1 3^1 + \binom{2}{2} 3^2$
Results in: $ t^2 + 6t + 9 $

The binomial theorem simplifies complex expansions, aiding deeper study in calculus.

Power Rule

The power rule is a fundamental principle in differentiation and integration. To integrate using the power rule, you follow this simple formula:

\[ \int t^n \,dt = \frac{t^{n+1}}{n+1} + C \]
Here are the key points:

$t$ is the variable.
$n$ is any real number except -1.
A constant of integration $C$ is added.

For example, with $t^{3/2}$:

Increase the exponent by 1: $\frac{3}{2} + 1 = \frac{5}{2}$.
Divide by the new exponent: $\frac{t^{5/2}}{\frac{5}{2}} = \frac{2}{5}t^{5/2}$.

Practice using the power rule with different values of $n$. This will help you become proficient in integrating polynomial functions.

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91影视

Evaluate. Each of the following can be integrated using the rules developed in this section, but some algebra may be required beforehand. $$ \int \frac{(t+3)^{2}}{\sqrt{t}} d t $$

Short Answer

Step by step solution

Expand the numerator

Rewrite the integral

Simplify the integrand

Integrate each term separately

Apply the power rule

Substitute back into the integral

Key Concepts

Definite Integral

Indefinite Integral

Binomial Theorem

Power Rule

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