Problem 9 Determine these indefinite integ... [FREE SOLUTION]

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Calculus and Its Applications

Marvin L. Bittinger, David J. Ellenbogen, Scott A. Surgent

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 4: Problem 9

Determine these indefinite integrals. $$\int\left(2 t^{2}+5 t-3\right) d t$$

Short Answer

Expert verified

\[\int(2t^{2}+5t-3)dt = \frac{2t^{3}}{3} + \frac{5t^{2}}{2} - 3t + C \]

Step by step solution

Identify the integrand

Recognize the function that needs to be integrated: $2 t^{2}+5 t-3$

Apply the Power Rule of Integration

For each term in the integrand, apply the power rule of integration, which states $\frac{d}{dx}\big(x^n\big) = \frac{x^{n+1}}{n+1} + C$.

Integrate each term separately

Integrate $2 t^{2}$, $5 t$, and $-3$ separately. The integrals are: $\frac{2t^{3}}{3}$, $\frac{5t^{2}}{2}$, and $-3t$.

Write the full integral with constants

Combine the integrated terms and add the constant of integration: $\frac{2t^{3}}{3} + \frac{5t^{2}}{2} - 3t + C$.

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Key Concepts

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

Power Rule of Integration

The power rule of integration is a fundamental technique in calculus. It helps us integrate functions by increasing the exponent of each term by one, and then dividing by that new exponent. For instance, if we have a term like $x^n$, its integral would be $\frac{x^{n+1}}{n+1} + C$.

Let's break down the power rule for better understanding:

Identify the exponent in the term: For $t^2$, the exponent is 2.
Increase the exponent by one: For $t^2$, it becomes $t^{2+1} = t^3$.
Divide by the new exponent: For $t^3$, it becomes $\frac{t^3}{3}$.

Applying the power rule helps make integration straightforward and systematic.

Integrand

An integrand is the function you want to integrate. In our example, the integrand is $2t^2 + 5t - 3$. Breaking down the integrand helps simplify the integration process because each term is handled separately.

Here鈥檚 how to deal with the integrand:

Focus on each term: Identify all individual terms, like $2t^2$, $5t$, and $-3$.
Apply rules separately: Integrate each term by applying the power rule.

By focusing on the integrand and breaking it down into manageable parts, integrating becomes efficient and less confusing.

Constant of Integration

The constant of integration, often denoted by $C$, is an essential part of indefinite integrals. When integrating a function, we always add this constant because differentiation of a constant is zero.

Here鈥檚 why the constant of integration is important:

Represents family of functions: Integrating can yield multiple functions that differ by a constant.
Ensures completeness: Adding $C$ ensures our integral represents all possible antiderivatives.

For example, integrating 2 gives $2t + C$, where $C$ could be any real number.

Integration of Polynomials

Integrating polynomials involves applying integration rules to each term of the polynomial. The power rule of integration is particularly helpful in this process.

For example, let鈥檚 look at the polynomial $2t^2 + 5t - 3$. We integrate each term separately:

Integrating $2t^2$ gives $\frac{2t^3}{3}$.
Integrating $5t$ gives $\frac{5t^2}{2}$.
Integrating $-3$ gives $-3t$.

Combining these results, we get $\frac{2t^3}{3} + \frac{5t^2}{2} - 3t + C$.

This method makes integrating complex polynomials manageable by breaking them into simpler, familiar parts.

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91影视

Determine these indefinite integrals. $$\int\left(2 t^{2}+5 t-3\right) d t$$

Short Answer

Step by step solution

Identify the integrand

Apply the Power Rule of Integration

Integrate each term separately

Write the full integral with constants

Key Concepts

Power Rule of Integration

Integrand

Constant of Integration

Integration of Polynomials

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