81. If [x] denotes the greatest integer function (for example, [1.16] = 1 and [1.8] = 1), then

∫₁^√3 (1 + [x])/(1 + x²) dx = L,

then L = ______ degrees.

Evaluate ∫₁^√3 (1 + [x])/(1 + x²) dx – Complete Concept and Detailed Solution

The key to solving these questions is to observe the interval of integration carefully. Since the greatest integer function remains constant over every interval between consecutive integers, the complicated-looking integrand often simplifies into an elementary function. Once this simplification is made, the remaining integration becomes straightforward.

Correct Answer

30°

Understanding the Greatest Integer Function

The greatest integer function, denoted by [x], returns the greatest integer less than or equal to x.

For example,

• [1.16] = 1
• [1.80] = 1
• [2.99] = 2
• [3] = 3

In the present question, the interval of integration is from 1 to √3.

Since

√3 ≈ 1.732,

every value of x in this interval satisfies

1 ≤ x < 2.

Therefore, throughout the entire interval,

[x] = 1.

Step 1: Simplify the Integrand

Since

[x] = 1,

the numerator becomes

1 + [x] = 1 + 1 = 2.

Hence, the integral simplifies to

∫₁^√3 2/(1+x²) dx.

The greatest integer function has now disappeared, leaving a standard calculus problem.

Step 2: Integrate

Recall the standard formula

∫ dx/(1+x²) = tan⁻¹x + C.

Therefore,

∫ 2/(1+x²) dx = 2 tan⁻¹x.

Applying the limits,

L = 2[tan⁻¹x]₁^√3

= 2(tan⁻¹√3 − tan⁻¹1).

Step 3: Substitute Standard Trigonometric Values

We know that

tan⁻¹(√3) = 60° = π/3

and

tan⁻¹(1) = 45° = π/4.

Hence,

L = 2(π/3 − π/4)

= 2(π/12)

= π/6.

Step 4: Convert into Degrees

Since

π radians = 180°,

we obtain

π/6 = 30°.

Therefore,

L = 30°.

Mathematical Verification

On the interval [1, √3],

[x] = 1 ✔

Integrand = 2/(1+x²) ✔

Integral = 2[tan⁻¹x]₁^√3 ✔

= 2(π/3 − π/4)

= π/6

= 30° ✔

The calculation is completely verified.

Related Practice Example

Evaluate

∫₂³ ([x]+1)/(1+x²) dx.

Since

2 ≤ x < 3,

[x] = 2.

Therefore,

the integral becomes

3∫₂³ dx/(1+x²),

which can be evaluated directly using the inverse tangent formula.

This follows exactly the same approach as the present question.

Key Takeaways

Whenever a definite integral contains the greatest integer function, first identify the interval of integration and determine the constant value of the floor function on that interval. After simplification, evaluate the resulting elementary integral using standard formulas. This approach greatly reduces computation and improves speed during competitive examinations.

Final Answer

Since [x]=1 throughout the interval [1,√3],

L = 2∫₁^√3 dx/(1+x²)

= π/6 = 30°.

Final Answer: 30°