https://file.aiquickdraw.com/mathbot/file/1726579379847pnpwrufa.jpg

The integral provided in the image is:

\[
I = \int_1^5 x \sqrt{x - 1} \, dx
\]

You requested a solution without using substitution, so let's proceed step by step using integration by parts.

### Step 1: Express the integral in a useful form.

We aim to use the formula for **integration by parts**:

\[
\int u \, dv = uv - \int v \, du
\]

Let's set up \( u \) and \( dv \) from the given integral \( I = \int_1^5 x \sqrt{x-1} \, dx \).

1. Choose \( u = x \), so that \( du = dx \).
2. Let \( dv = \sqrt{x - 1} \, dx \), and now we need to find \( v \).

### Step 2: Find \( v \).

We know that:

\[
dv = \sqrt{x - 1} \, dx = (x - 1)^{1/2} \, dx
\]

To integrate this, use the power rule:

\[
v = \int (x - 1)^{1

Solve the integral: I = ∫₁⁵ x√(x - 1) dx without using substitution.

This detailed guide explains how to solve the integral ∫₁⁵ x√(x - 1) dx using integration by parts, step by step. Learn how to apply integration techniques without substitution, suitable for undergraduate calculus students.

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