Let's break down the given integral using u-substitution. The integral is:

$\int_1^4 (5x - 3)^5 \, dx$

We are asked to apply the method of u-substitution where the general form is:

$\int_a^b f(g(x))g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du$

Step 1: Choosing $u$

To simplify the integral, we let: $u = 5x - 3$

Step 2: Differentiating $u$

Next, we differentiate $u$ with respect to $x$ to find $du$:

$du = 5 \, dx \quad \text{or equivalently} \quad dx = \frac{du}{5}$

Step 3: Changing the limits of integration

The original limits of integration are from $x = 1$ to $x = 4$. We need to convert these limits in terms of $u$.

• When $x = 1$: $u = 5(1) - 3 = 2$
• When $x = 4$: $u = 5(4) - 3 = 17$

So the limits of integration for $u$ are from 2 to 17.

Step 4: Substituting into the integral

The integral becomes:

$\int_2^{17} u^5 \cdot \frac{du}{5}$

Simplifying:

$\frac{1}{5} \int_2^{17} u^5 \, du$

Step 5: Solving the integral

Now we integrate $u^5$:

$\int u^5 \, du = \frac{u^6}{6}$

Thus, the integral becomes:

$\frac{1}{5} \cdot \left[ \frac{u^6}{6} \right]_2^{17}$

Step 6: Applying the limits

Substitute $u = 17$ and $u = 2$:

$\frac{1}{5} \cdot \left( \frac{17^6}{6} - \frac{2^6}{6} \right)$

Now, let's compute the result. I will calculate this.The value of the original integral is:

$804583.5$

Here’s a summary of the u-substitution steps:

• $u = 5x - 3$
• $du = 5 \, dx$, so $dx = \frac{du}{5}$
• Limits for $u$ go from 2 to 17
• The integral simplifies to $\frac{1}{5} \int_2^{17} u^5 \, du$
• The computed result is $804583.5$

Would you like further clarification on any of the steps, or do you have additional questions?

5 Related Questions:

1. How does u-substitution simplify complex integrals?
2. Why do we need to adjust the limits when performing u-substitution?
3. Can u-substitution be applied to any integral?
4. What happens if you forget to change the limits during substitution?
5. What is the significance of the constant factor when substituting $du$?

Tip:

Always check if you need to adjust the limits of integration after making a substitution. This is a common point where errors can occur in definite integrals.