Problem 14 - Applied Calculus

Question

Find the integrals .Check your answers by differentiation. $$\int t^{2}\left(t^{3}-3\right)^{10} d t$$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{33} (t^3 - 3)^{11} + C \).

Step 1: Identify the problem type

The given problem is to find the integral of \( t^2 (t^3 - 3)^{10} \) with respect to \( t \). This suggests a substitution method because of the composition of functions.

Step 2: Choose the substitution

We choose \( u = t^3 - 3 \) since it simplifies the expression inside the brackets raised to a power. This choice simplifies the integral due to the presence of \( t^2 \) in the original integrand.

Step 3: Differentiate the substitution

Differentiate \( u = t^3 - 3 \) with respect to \( t \) to find \( du/dt \). This gives us \( \frac{du}{dt} = 3t^2 \), so \( du = 3t^2 \, dt \).

Step 4: Solve for dt

From \( du = 3t^2 \, dt \), we can solve for \( dt \) as follows: \( dt = \frac{du}{3t^2} \).

Step 5: Substitute into the integral

Now substitute into the integral: \[ \int t^2 (t^3 - 3)^{10} \, dt = \int t^2 u^{10} \cdot \frac{du}{3t^2} \]. The \( t^2 \) terms cancel, leaving: \[ \frac{1}{3} \int u^{10} \, du \].

Step 6: Integrate

Integrate \( \frac{1}{3} \int u^{10} \, du \). This gives \( \frac{1}{3} \cdot \frac{u^{11}}{11} + C = \frac{1}{33} u^{11} + C \).

Step 7: Back-substitute \( u \)

Replace \( u \) with \( t^3 - 3 \) to revert to the variable \( t \). The integral becomes: \( \frac{1}{33} (t^3 - 3)^{11} + C \).

Step 8: Differentiate to verify

Differentiate \( \frac{1}{33} (t^3 - 3)^{11} + C \) with respect to \( t \) to check our work. Applying the chain rule, the derivative should match the original integrand \( t^2 (t^3 - 3)^{10} \). Verify the calculation to ensure they are the same.

Key Concepts

Substitution Method

When handling integrals where a composition of functions is involved, the substitution method becomes a powerful tool. In this specific exercise, the core of the expression inside the integral is the function

• \((t^3 - 3)^{10}\)

By choosing an appropriate substitution, the integral can be simplified. Here, we select:

• \(u = t^3 - 3\)

The goal is to change variables, simplifying the integral into a more manageable form. This substitution transforms the original function, making it easier to integrate.

Furthermore, after substituting, the problem converts to integrating

• \(\int u^{10} \, du\)

By substituting back to the original variable after integrating, we complete the process.

Chain Rule

The chain rule is a fundamental concept in calculus, particularly useful for differentiating compositions of functions. When tackling the differentiation step to verify our answer, the chain rule assists in tracing how changes in the inner function affect the overall expression.

In the solution:

• The function to differentiate was \( (t^3 - 3)^{11} \)
• The chain rule states that for a composite function \( f(g(x)) \), its derivative is \( f'(g(x)) \cdot g'(x) \)

So, applying this rule, the derivative of \((t^3 - 3)^{11}\) simplifies back to the integrand \( t^2 (t^3 - 3)^{10} \), confirming our original answer by linking back to the original problem.

Differentiation

Differentiation is the mathematical process of finding the derivative of a function. It allows us to verify our integration result was correct. Throughout the exercise:

• After integrating to get \( \frac{1}{33}(t^3 - 3)^{11} + C \), we use differentiation to check our work
• Using the chain rule, the derivative matches the original integrand, verifying correctness

Differentiation confirms the solution to an integral by checking that differentiating the result returns us to the initial function. This step ensures accountability and clarity, guarding against calculation errors.

Integrand Simplification

Simplifying the integrand is a crucial initial step when tackling complex integrals. Here, by recognizing parts of the expression that can simplify:

• The term \( t^2 (t^3 - 3)^{10} \) translates to an integral involving \( u^{10} \) after substitution

The cancellation of terms during substitution often reveals simpler integrands, reducing unnecessary computational complexity.

The main advantage of simplifying is that it transforms a daunting expression into one that's straightforward and computationally friendly, easing the integration process significantly. This practice is often the secret to efficiently solving integration problems.