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Let's solve the integral step by step, using the given substitution \( u = 1 - 6t \).

### Problem:
\[
\int \frac{dt}{(1 - 6t)^4}
\]
Given: \( u = 1 - 6t \)

### Step 1: Substitution
We are given \( u = 1 - 6t \). Now, differentiate \( u \) with respect to \( t \):
\[
du = -6 \, dt \quad \Rightarrow \quad dt = -\frac{du}{6}
\]

### Step 2: Rewrite the integral
Substitute \( u \) and \( dt \) into the integral:
\[
\int \frac{dt}{(1 - 6t)^4} = \int \frac{-\frac{du}{6}}{u^4}
\]
This simplifies to:
\[
-\frac{1}{6} \int u^{-4} \, du
\]

### Step 3: Integrate
Now, integrate \( u^{-4} \):
\[
\int u^{-4} \, du = \frac{u^{-3}}{-3} = -\frac{1}{3u^3}
\]

### Step 4: Multiply by the constant
Substitute back into the expression:
\[
-\frac{1}{6} \cdot -\frac{1}{3u^3} = \frac{1}{18u^3}
\]

### Step 5: Substitute \( u = 1 - 6t \) back
Finally, substitute \( u = 1 - 6t \) back into the result:
\[
\frac{1}{18(1 - 6t)^3} + C
\]

### Final Answer:
\[
\int \frac{dt}{(1 - 6t)^4} = \frac{1}{18(1 - 6t)^3} + C
\]

Would you like more details, or do you have any questions about the process?

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Here are 5 related questions you might want to explore:
1. How do you apply integration by substitution in more complex integrals?
2. How would this integral change if the exponent was different, for example, \( (1 - 6t)^n \)?
3. What happens if we change the substitution to a different variable?
4. Can you solve a similar integral but with \( (1 - 6t)^2 \) in the denominator?
5. How would this integral look with bounds for definite integration?

**Tip:** Always double-check your substitution to ensure that the transformation is correct before proceeding with the integration.

Can you solve the integral \( \int \frac{dt}{(1 - 6t)^4} \) with the substitution \( u = 1 - 6t \)?

Learn how to solve the integral \( \int \frac{dt}{(1 - 6t)^4} \) step-by-step using the substitution method. This solution includes substitution formulas and the power rule for integration, suitable for calculus students in Grades 11-12 or early college-level courses.

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