Problem 26 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int t^{2}\left(t^{3}-1\right)^{7} d t $$

Step-by-Step Solution

Answer

\[ \int t^2 (t^3 - 1)^7 \, dt = \frac{1}{24} (t^3 - 1)^8 + C \]

Step 1: Identify the Integral Type

The given integral is \( \int t^2 (t^3 - 1)^7 \, dt \). It seems well-suited for a substitution method, considering the composite function \( (t^3 - 1)^7 \).

Step 2: Choose a Substitution

Let \( u = t^3 - 1 \). Then the differential \( du = 3t^2 \, dt \). We can solve for \( dt \) in terms of \( du \): \( dt = \frac{du}{3t^2} \).

Step 3: Substitute and Adjust the Integral

Substitute \( u = t^3 - 1 \) into the integral, and replace \( t^2 \, dt \) using the expression from the substitution: \[ \int t^2 (t^3 - 1)^7 \, dt = \int t^2 u^7 \cdot \frac{du}{3t^2} = \frac{1}{3} \int u^7 \, du. \]

Step 4: Integrate with Respect to \( u \)

The integral \( \frac{1}{3} \int u^7 \, du \) becomes: \[ \frac{1}{3} \cdot \frac{u^8}{8} = \frac{1}{24} u^8 + C. \]

Step 5: Substitute Back the Original Variable

Replace \( u = t^3 - 1 \) back into the expression: \[ \frac{1}{24} (t^3 - 1)^8 + C. \]

Step 6: Differentiate to Verify the Solution

Verify the integral by differentiating \( \frac{1}{24} (t^3 - 1)^8 \). Using the chain rule: \[ \frac{d}{dt} \left( \frac{1}{24} (t^3 - 1)^8 \right) = \frac{1}{24} \cdot 8 (t^3 - 1)^7 \cdot 3t^2 = t^2 (t^3 - 1)^7. \] Hence, the differentiation confirms the solution is correct.

Key Concepts

Substitution Method

The substitution method is a powerful technique used in calculus for simplifying the process of integration. When you encounter an integral with a composite function, like the one in this exercise, it often supports the use of substitution. The essence of the method involves swapping a part of the integrand with a new variable. This simplification transforms the integrand into a form that is much easier to handle.

To apply substitution:

• Identify a part of the integrand that, when substituted, simplifies the expression. In this case, we chose \( u = t^3 - 1 \).
• Find the differential of your substituted variable. Here, we obtained \( du = 3t^2 \, dt \).
• Re-arrange this to express \( dt \) in terms of \( du \). For our exercise, \( dt = \frac{du}{3t^2} \).
• Replace the identified portions in the original integral with the new variables and expressions. In this example: \( t^2 (t^3 - 1)^7 \, dt \) became \( \frac{1}{3} \int u^7 \, du \).

By integrating with respect to \( u \), the calculation becomes straightforward.

Definite and Indefinite Integrals

Integrals come in two types: definite and indefinite. This exercise focuses on an indefinite integral, which means we are finding an antiderivative of the function. The result will contain a constant of integration, \( C \), signifying the family of functions that differ by a constant.

Indefinite integrals:

• Are expressed with the integral sign \( \int \) without upper and lower limits.
• Include an arbitrary constant, \( C \), because the derivative of a constant is zero, making it indeterminate from the derivative alone.

In contrast, definite integrals have limits, providing a numeric result once evaluated. Here, we focus on indefinite integration. After substituting and simplifying our integral, we calculated \( \int u^7 \, du \) to obtain \( \frac{u^8}{8} + C \), which translates back to \( \frac{1}{24} (t^3 - 1)^8 + C \) when reverting to the original variable.

Verification by Differentiation

Verification by differentiation ensures the correctness of our integration process. Once you've derived an antiderivative, it's crucial to differentiate your result to see if you return to the original function.

The general steps include:

• Differentiating your final antiderivative expression, which in this exercise is \( \frac{1}{24} (t^3 - 1)^8 + C \).
• Applying the chain rule for composite functions, illustrated as \( \frac{d}{dt}\big(\frac{1}{24} (t^3 - 1)^8\big) = \frac{1}{24} \cdot 8 (t^3 - 1)^7 \cdot\frac{d}{dt}(t^3 - 1) \).
• Conducting algebraic simplification after differentiation to verify it matches the original integrand.

In our case, differentiating yields \( t^2 (t^3 - 1)^7 \), confirming the antiderivative is correct. This verification step builds confidence in our integration skills by solidifying our approach with backward validation.