Certainly! Let’s evaluate the definite integral [tex]\(\int_0^1 \cos (3x + 5) \, dx\)[/tex].
### Step 1: Variable Substitution
First, we'll use a substitution to simplify the integral. Let's set:
[tex]\[ u = 3x + 5 \][/tex]
Thus, the differential [tex]\( du \)[/tex] is:
[tex]\[ du = 3dx \quad \text{or} \quad dx = \frac{du}{3} \][/tex]
### Step 2: Adjust the Limits of Integration
Next, we need to change the limits of integration according to the substitution [tex]\( u = 3x + 5 \)[/tex].
When [tex]\( x = 0 \)[/tex]:
[tex]\[ u = 3(0) + 5 = 5 \][/tex]
When [tex]\( x = 1 \)[/tex]:
[tex]\[ u = 3(1) + 5 = 8 \][/tex]
So the limits of integration change from [tex]\( x \in [0, 1] \)[/tex] to [tex]\( u \in [5, 8] \)[/tex].
### Step 3: Rewrite the Integral
The integral [tex]\(\int_0^1 \cos(3x + 5) \, dx\)[/tex] in terms of [tex]\( u \)[/tex] is:
[tex]\[ \int_5^8 \cos(u) \cdot \frac{1}{3} \, du \][/tex]
### Step 4: Integrate
We can factor out the constant [tex]\(\frac{1}{3}\)[/tex] from the integral:
[tex]\[ \frac{1}{3} \int_5^8 \cos(u) \, du \][/tex]
The antiderivative of [tex]\(\cos(u)\)[/tex] is [tex]\(\sin(u)\)[/tex], hence we have:
[tex]\[ \frac{1}{3} \left[ \sin(u) \right]_5^8 \][/tex]
### Step 5: Evaluate the Antiderivative at the Bounds
Now we substitute the limits of integration into [tex]\(\sin(u)\)[/tex]:
[tex]\[ \frac{1}{3} \left( \sin(8) - \sin(5) \right) \][/tex]
### Final Answer
Thus, the value of the definite integral is:
[tex]\[ \int_0^1 \cos(3x + 5) \, dx = \frac{\sin(8) - \sin(5)}{3} \][/tex]
or equivalently,
[tex]\[ \int_0^1 \cos(3x + 5) \, dx = \frac{\sin(8)}{3} - \frac{\sin(5)}{3} \][/tex]
which simplifies to:
[tex]\[ -\frac{\sin(5)}{3} + \frac{\sin(8)}{3} \][/tex]
This is the evaluated result of the definite integral.
