Integration Worksheet - Substitution Method Solutions

Let’s solve problem #3 from the worksheet:

Problem 3:
∫ (4x − 5) dx

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We’ll use substitution, just like in problems 1 and 2.

Step 1: Choose a substitution

Look at the inside of the square root: 4x − 5.
Let’s let:
> u = 4x − 5

Step 2: Find du

Take the derivative of u with respect to x:
du/dx = 4 → so,
> du = 4 dx

But we need dx by itself for substitution. So divide both sides by 4:
> dx = (1/4) du

Step 3: Substitute into the integral

Original integral:
∫ √(4x − 5) dx

Replace (4x − 5) with u, and replace dx with (1/4) du:
= ∫ √u · (1/4) du

We can pull the constant (1/4) out front:
= (1/4) ∫ √u du

Now, rewrite √u as a power:
√u = u^(1/2)

So now:
= (1/4) ∫ u^(1/2) du

Step 4: Integrate

Use the power rule for integration:
∫ u^n du = (u^(n+1)) / (n+1) + C, as long as n ≠ -1

Here, n = 1/2, so:
∫ u^(1/2) du = u^(3/2) / (3/2) = (2/3) u^(3/2)

Multiply by the 1/4 we had outside:
= (1/4) · (2/3) u^(3/2) + C
= (2/12) u^(3/2) + C
= (1/6) u^(3/2) + C

Step 5: Substitute back

Remember, u = 4x − 5, so replace u:
= (1/6)(4x − 5)^(3/2) + C

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Final Answer:
(1/6)(4x − 5)^(3/2) + C