Questions: Use the substitution u=5x+7 to evaluate the integral ∫ sin(5x+7) dx.

Transcript text: Use the substitution $u=5 x+7$ to evaluate the integral $\int \sin (5 x+7) d x$.

Solution

Solution Steps

To evaluate the integral using substitution, we will let $ u = 5x + 7 $. Then, we find the derivative $ du = 5 \, dx $, which implies $ dx = \frac{1}{5} \, du $. Substitute these into the integral to transform it into an integral in terms of $ u $.

Step 1: Substitution

We start with the integral $ \int \sin(5x + 7) \, dx $. We use the substitution $ u = 5x + 7 $, which gives us $ du = 5 \, dx $ or $ dx = \frac{1}{5} \, du $.

Step 2: Transforming the Integral

Substituting $ u $ into the integral, we have: \[ \int \sin(5x + 7) \, dx = \int \sin(u) \cdot \frac{1}{5} \, du = \frac{1}{5} \int \sin(u) \, du \]

Step 3: Evaluating the Integral

The integral of $ \sin(u) $ is $ -\cos(u) $. Therefore, we have: \[ \frac{1}{5} \int \sin(u) \, du = \frac{1}{5} \cdot (-\cos(u)) = -\frac{1}{5} \cos(u) \]

Step 4: Back Substitution

Now, we substitute back $ u = 5x + 7 $: \[ -\frac{1}{5} \cos(5x + 7) + C \]