Problem 6 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int\left(2 t^{5}-3\right) t^{4} d t $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{t^{10}}{5} - \frac{3t^5}{5} + C \), confirmed by differentiation.

Step 1: Expand the Expression

Distribute the term \( t^4 \) across \( 2t^5 - 3 \) within the integral. This means you rewrite the integrand as \( 2t^5 \cdot t^4 - 3t^4 \). This simplifies to \( 2t^9 - 3t^4 \). The integral is now \( \int (2t^9 - 3t^4) \, dt \).

Step 2: Integrate Each Term Separately

Integrate each term individually:- For \( 2t^9 \), apply the power rule \( \int t^n \, dt = \frac{t^{n+1}}{n+1} \). So, \( \int 2t^9 \, dt = \frac{2t^{10}}{10} = \frac{t^{10}}{5} \).- For \( -3t^4 \), integrate using the power rule: \( \int -3t^4 \, dt = -3 \cdot \frac{t^{5}}{5} = -\frac{3t^5}{5} \).

Step 3: Combine Integrated Terms

Add the results of each individual integration to find the indefinite integral:\[ \int (2t^9 - 3t^4) \, dt = \frac{t^{10}}{5} - \frac{3t^5}{5} + C \]Where \( C \) is the constant of integration.

Step 4: Differentiate to Verify

Differentiate the expression \( \frac{t^{10}}{5} - \frac{3t^5}{5} + C \) to verify correctness:- The derivative of \( \frac{t^{10}}{5} \) is \( \frac{10t^9}{5} = 2t^9 \).- The derivative of \( -\frac{3t^5}{5} \) is \( -\frac{15t^4}{5} = -3t^4 \).- The derivative of constant \( C \) is zero.Combining these derivatives, you should return to the original integrand: \( 2t^9 - 3t^4 \).This confirms the integration was performed correctly.

Key Concepts

Power Rule

The power rule is a fundamental tool in both differentiation and integration, making it essential in calculus. It helps in finding antiderivatives when working with polynomial terms. When using the power rule for integration, the general form to keep in mind is:\[\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\]Here’s how it works:

• Add 1 to the original exponent.
• Divide the term by the new exponent.

Remember, this rule only applies when the exponent, \( n \), is not equal to -1, because division by zero isn't allowed. In this exercise, we applied the power rule to each term separately:- For \( 2t^9 \), we get \( \frac{t^{10}}{5} \).- For \(-3t^4\), it resulted in \(-\frac{3t^5}{5}\).These terms were combined to create the antiderivative of the entire expression.

Differentiation

Differentiation is the process of finding the derivative of a function, which essentially calculates the rate of change. It's the inverse operation of integration. Here, we used differentiation to verify the accuracy of our integration by performing the following steps:

• Take the derivative of each term separately.
• The derivative of \( \frac{t^{10}}{5} \) is \( 2t^9 \).
• The derivative of \(-\frac{3t^5}{5}\) is \(-3t^4\).
• The derivative of the constant \( C \) is zero.

Combining all these, we returned to the original expression \(2t^9 - 3t^4\), confirming that our integration was correct. Differentiation helps ensure we haven't made mistakes in our calculations.

Constant of Integration

The constant of integration, represented by \( C \), is a vital component in indefinite integrals. It accounts for the fact that antiderivatives are determined up to an additive constant because differentiation of any constant yields zero. Consequently, different functions might return to a single derivative when you subtract the constant's derivative.Here are some key points about the constant of integration:

• Only appears in indefinite integrals – these integrals have no specific limits.
• Reflects the family of functions that differ by a constant, which share the same derivative.
• Doesn't affect the derivative of the function as it's derivative is always zero.

In our problem, \( C \) ensures that the solution represents all possible antiderivatives of the function \( 2t^9 - 3t^4 \). Understanding this concept is crucial when solving indefinite integrals.