The value of _0^π /2dx1+tan^3x is

C. π /4. Step 1: Write the integrand as cos^3xsin^3x+cos^3x and call the integral I. Step 2: Replacing x by π /2-x gives I=_0^π /2sin^3xsin^3x+cos^3x dx. Step 3: Adding the two forms: 2I=_0^π /21 dx=π /2, so I=π /4. Check: The integrand is 1 at x=0, 1/2 at x=π /4 and 0 at x=π /2, so its average value is 1/2 and I=1/2× π /2. Trap: Forgetting to halve 2I gives π /2, option (D).

Question 19 of 31 · Definite Integration

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NIMCET 2008

The value of $\int_{0}^{π /2} \frac{d x}{1 + t an ^{3} x}$ is

Answer (Detailed Solution Below)View answer ▾

Answer: Option C — $\frac{π}{4}$

Detailed Solution

Write the integrand as $\frac{cos ^{3} x}{s in ^{3} x + cos ^{3} x}$ and call the integral $I$ .

Replacing $x$ by $\frac{π}{2} - x$ gives $I = \int_{0}^{π /2} \frac{s in ^{3} x}{s in ^{3} x + cos ^{3} x} d x$ .

Adding the two forms: $2 I = \int_{0}^{π /2} 1 d x = \frac{π}{2}$ , so $I = \frac{π}{4}$ .

✓ Check

The integrand is

1

x = 0

\frac{1}{2}

x = \frac{π}{4}

and

0

x = \frac{π}{2}

, so its average value is

\frac{1}{2}

and

I = \frac{1}{2} \times \frac{π}{2}

⚠ Common trap

Forgetting to halve

2 I

gives

\frac{π}{2}

, option (D).

Quick tip

$\int_{0}^{π /2} \frac{d x}{1 + t a n ^{n} x} = \frac{π}{4}$ for every n

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Definite Integration — Preparation

This question tests your understanding of Definite Integration, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A definite integral gives a number rather than a family of functions. It is evaluated using the fundamental theorem of calculus: the integral of f(x) from a to b equals F(b) - F(a), where F is an antiderivative of f. This topic covers evaluating definite integrals, substitution with a change of limits, integration by parts with limits, and the definite integral as the limit of a sum. The properties of definite integrals are the key to most exam questions. The most useful is the "a + b - x" (or "a - x") property, which turns many difficult integrals into simple ones by adding the integral to itself. Also remember that the integral of an odd function over a symmetric interval from -a to a is zero, while the integral of an even function equals twice the integral from 0 to a. Practise the definite integration questions below, each with a step-by-step solution showing which property is used.

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