91. Evaluate the definite integral
∫0π |cos x| dx.
Evaluate the Integral ∫₀π |cos x| dx – Detailed Solution
Understanding the Problem
This question belongs to the topic of Definite Integration and tests the concept of integrating functions containing the modulus (absolute value). Whenever an integral contains an absolute value, the first step is to identify the intervals where the function inside the modulus is positive and where it is negative. The modulus changes the sign of the function wherever it becomes negative, so the integral must be evaluated piecewise.
For the function cos x, the sign changes at x = π/2 in the interval [0, π]. Therefore, the given integral should be divided into two simpler integrals.
Step 1: Determine the Sign of cos x
On the interval
0 ≤ x ≤ π/2,
cos x ≥ 0.
Hence,
|cos x| = cos x.
On the interval
π/2 ≤ x ≤ π,
cos x ≤ 0.
Therefore,
|cos x| = -cos x.
Step 2: Split the Integral
Using the above property, the given integral becomes
∫0π |cos x| dx = ∫0π/2 cos x dx + ∫π/2π (-cos x) dx.
Step 3: Evaluate the First Integral
Since
∫ cos x dx = sin x,
we have
∫0π/2 cos x dx = [sin x]0π/2
= sin(π/2) − sin(0)
= 1 − 0
= 1.
Step 4: Evaluate the Second Integral
Now evaluate
∫π/2π (-cos x) dx.
The integral of -cos x is
-sin x.
Therefore,
∫π/2π (-cos x) dx = [-sin x]π/2π
= -sin(π) + sin(π/2)
= 0 + 1
= 1.
Step 5: Add Both Results
The required integral is
1 + 1 = 2.
Final Answer
∫0π |cos x| dx = 2.
Why This Method Works
The modulus function always returns a non-negative value. Whenever the expression inside the modulus changes sign, the interval of integration must be divided accordingly. Since cos x is positive on the first half of the interval and negative on the second half, replacing |cos x| with the correct expression on each interval converts the problem into two standard definite integrals that are easy to evaluate.
Key Concepts Covered
Modulus Function
For any real-valued function f(x),
|f(x)| = f(x) when f(x) ≥ 0,
and
|f(x)| = -f(x) when f(x) < 0.
Sign of cos x
In the interval [0, π], the cosine function is positive on [0, π/2] and negative on [π/2, π]. This change in sign is the key to solving the problem.
Definite Integration
When integrating piecewise-defined functions, each interval is evaluated separately, and the individual results are added to obtain the final value of the definite integral.
Alternative Short Method
The graph of |cos x| is perfectly symmetric about x = π/2. Therefore,
∫0π |cos x| dx = 2∫0π/2 cos x dx
= 2 × 1
= 2.
This symmetry-based approach is often the fastest method in competitive examinations.
Correct Answer: 2
