Problem 20 - Calculus Single Variable

Question

In Exercises \(11-20,\) calculate \(F(x)=\int_{a}^{x} f(t) d t\) $$ f(t)=(3 t+1 / 2) / \sqrt{t} \quad a=4 \quad(0

Step-by-Step Solution

Answer

The integral is \( F(x) = 2x^{3/2} + x^{1/2} - 18 \).

Step 1: Understand the Function to Integrate

We're given the function \( f(t) = \frac{3t + \frac{1}{2}}{\sqrt{t}} \). To integrate it, first simplify if possible. The function combines polynomials and a square root.

Step 2: Simplify the Integrand

Simplify \( f(t) = \frac{3t + \frac{1}{2}}{\sqrt{t}} \) by splitting the fraction:\[\frac{3t}{\sqrt{t}} + \frac{\frac{1}{2}}{\sqrt{t}} \Rightarrow 3t^{1/2} + \frac{1}{2} t^{-1/2}\]

Step 3: Integrate the Simplified Function

Apply the integral \( F(x) = \int_{4}^{x} \left( 3t^{1/2} + \frac{1}{2} t^{-1/2} \right) dt \). Calculate each part separately:- For \( 3t^{1/2} \), use the power rule for integration: \[ \int 3t^{1/2} \, dt = 3 \cdot \frac{2}{3}t^{3/2} = 2t^{3/2} \]- For \( \frac{1}{2}t^{-1/2} \), use the power rule:\[ \int \frac{1}{2}t^{-1/2} \, dt = \frac{1}{2} \cdot 2t^{1/2} = t^{1/2} \]

Step 4: Evaluate the Definite Integral

Now, evaluate the definite integral to find \( F(x) \):\[F(x) = \left[ 2t^{3/2} + t^{1/2} \right]_{t=4}^{t=x}\]Substitute the bounds:\[= \left( 2x^{3/2} + x^{1/2} \right) - \left( 2(4)^{3/2} + (4)^{1/2} \right)\]Calculate the values:- \(2(4)^{3/2} = 2 \times 8 = 16\)- \((4)^{1/2} = 2\)So, the definite integral becomes:\[F(x) = 2x^{3/2} + x^{1/2} - 18\]

Step 5: Final Answer

The final expression after integrating and evaluating the definite integral is: \[ F(x) = 2x^{3/2} + x^{1/2} - 18 \]

Key Concepts

Function Simplification

When solving integrals, simplifying the function first can make integration much easier. A complex function like \( f(t) = \frac{3t + \frac{1}{2}}{\sqrt{t}} \) can seem daunting, but breaking it down into simpler parts helps. Splitting the terms over the square root yields useful forms. To simplify:

• Break the fraction: \( \frac{3t}{\sqrt{t}} + \frac{\frac{1}{2}}{\sqrt{t}} \).
• This results in: \( 3t^{1/2} + \frac{1}{2} t^{-1/2} \).

This step transforms the function into terms that are power functions, which can be easily integrated using standard rules. By addressing such transformations, we simplify the process and make complex integrals manageable.

Power Rule in Integration

The power rule is a fundamental technique in calculus for integrating functions of the form \( t^n \). This rule states that for any real number \( n eq -1 \), the integral of \( t^n \) is:\[\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\]It simplifies the integration process, especially when dealing with functions composed of polynomials or simpler powers.For \( f(t) = 3t^{1/2} + \frac{1}{2} t^{-1/2} \), apply the power rule separately:

• \( \int 3t^{1/2} \, dt = 3 \times \frac{2}{3} t^{3/2} = 2t^{3/2} \)
• \( \int \frac{1}{2}t^{-1/2} \, dt = \frac{1}{2} \times 2t^{1/2} = t^{1/2} \)

Through understanding and applying the power rule, the function is integrated with ease into a more straightforward form to evaluate further.

Evaluating Definite Integrals

A definite integral, denoted as \( \int_{a}^{b} f(t) \, dt \), represents the signed area between the function \( f(t) \) and the \( t \)-axis, from \( t=a \) to \( t=b \). Evaluating a definite integral involves calculating the antiderivative and then applying the Fundamental Theorem of Calculus.After integrating the function, use this result:\[ F(x) = \left[ 2t^{3/2} + t^{1/2} \right]_{t=4}^{t=x} \]Apply the evaluated antiderivative at the interval's boundaries:

• Calculate at the upper limit \( x \): \( 2x^{3/2} + x^{1/2} \)
• Subtract the calculation at the lower limit \( 4 \): \( 2(4)^{3/2} + (4)^{1/2} = 16 + 2 = 18 \)

This methodology ensures that you find the net area under the curve, resulting in the final evaluated integral.

Integral Calculation Steps

Effectively calculating a definite integral involves a series of systematic steps:1. **Simplify the Function:** - Break down complex expressions into simpler parts, like transforming \( \frac{3t + \frac{1}{2}}{\sqrt{t}} \) into \( 3t^{1/2} + \frac{1}{2} t^{-1/2} \).2. **Integrate Using Known Rules:** - Apply the power rule to each term separately: - \( \int 3t^{1/2} \, dt = 2t^{3/2} \) - \( \int \frac{1}{2}t^{-1/2} \, dt = t^{1/2} \)3. **Set Up the Definite Integral:** - Write down the antiderivative: \[ F(x) = \left[ 2t^{3/2} + t^{1/2} \right]_{t=4}^{t=x} \]4. **Evaluate the Boundaries:** - Substitute the limits and subtract: - Final expression: \( F(x) = 2x^{3/2} + x^{1/2} - 18 \)Following these steps ensures accuracy and helps you systematically solve integrals without missing crucial parts.