∫_(π/4)^(3π/4) dx/(1 + cos x) is equal to (a) 2 (b) -2 (c) 1/2 (d) -1/2

Q: Wait, why do we multiply by $1 - \cos x$? Is there no other way?

That is a fantastic question! Multiplying by the conjugate is a standard 'trick' to simplify trigonometric denominators, but it is not the only way. You could also use the half-angle identity: $1 + \cos x = 2 \cos^2(x/2)$. This would transform the integral into $\int \frac{1}{2} \sec^2(x/2) dx$, which is also very straightforward to solve. Both methods are elegant and correct—mathematics often gives us multiple paths to the same beautiful truth!

Q: I keep getting confused with the negative signs when integrating $\csc^2 x$ and $\csc x \cot x$. Any tips?

It is completely normal to feel that way! The best way to avoid this trap is to write out the derivatives you know: $\frac{d}{dx}(\cot x) = -\csc^2 x$ and $\frac{d}{dx}(\csc x) = -\csc x \cot x$. Because the derivatives have negative signs, the integrals must also carry them. When you see $\int (\csc^2 x - \csc x \cot x) dx$, just handle each term carefully: $(-\cot x) - (-\csc x)$. The double negative becomes a positive. Slow down, write it out, and you will never miss it again!

Q: Why do we evaluate the limits at the end? Can we change the variable?

You absolutely can! If you prefer, you could use a substitution like $u = x/2$ or other trigonometric substitutions, but you would then need to change your limits of integration accordingly. However, for this specific problem, keeping the variable as $x$ and using the standard integral forms is often faster and less prone to calculation errors. Stick with what feels most intuitive to you!

Question

$\int_{\pi/4}^{3\pi/4} \frac{dx}{1 + \cos x}$ is equal to

Options:

(A) 2

(B) -2

(C) 1/2

(D) -1/2

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a complex integral: $I = \int_{\pi/4}^{3\pi/4} \frac{dx}{1 + \cos x}$. At first glance, it looks like a wall.

The denominator $1 + \cos x$ prevents us from using simple power rules or direct substitution. But in the world of JEE Advanced, every wall is just a door waiting for the right key.

### The Conjugate Key

The first step in our journey is to simplify the denominator. We have $1 + \cos x$.

If we multiply the numerator and the denominator by the conjugate, $1 - \cos x$, we create a beautiful opportunity. Because $(1 + \cos x)(1 - \cos x)$ is a difference of squares, it simplifies to $1 - \cos^2 x$.

We know, from the very foundation of trigonometry, that $1 - \cos^2 x = \sin^2 x$. Our integral now looks like this:

$$I = \int_{\pi/4}^{3\pi/4} \frac{1 - \cos x}{\sin^2 x} dx$$

### The Art of Splitting

Now, we have a single term in the denominator, $\sin^2 x$. This is a massive improvement! We can split the fraction into two distinct parts:

$$I = \int_{\pi/4}^{3\pi/4} \left( \frac{1}{\sin^2 x} - \frac{\cos x}{\sin^2 x} \right) dx$$

Look at these two terms. They are the building blocks of calculus. The first term, $\frac{1}{\sin^2 x}$, is simply $\csc^2 x$.

The second term, $\frac{\cos x}{\sin^2 x}$, can be rewritten as $\frac{1}{\sin x} \cdot \frac{\cos x}{\sin x}$, which is $\csc x \cot x$.

### The Final Integration

We have arrived at the heart of the problem:

$$I = \int_{\pi/4}^{3\pi/4} (\csc^2 x - \csc x \cot x) dx$$

We know that the derivative of $\cot x$ is $-\csc^2 x$, so the integral of $\csc^2 x$ is $-\cot x$. Similarly, the derivative of $\csc x$ is $-\csc x \cot x$, so the integral of $\csc x \cot x$ is $-\csc x$.

Putting it all together, we get:

$$I = \left[ -\cot x - (-\csc x) \right]_{\pi/4}^{3\pi/4} = \left[ \csc x - \cot x \right]_{\pi/4}^{3\pi/4}$$

### The Evaluation

Now, we just need to plug in our limits. For the upper limit, $x = 3\pi/4$: $\csc(3\pi/4) = \sqrt{2}$ and $\cot(3\pi/4) = -1$.

So, the upper limit value is $\sqrt{2} - (-1) = \sqrt{2} + 1$. For the lower limit, $x = \pi/4$: $\csc(\pi/4) = \sqrt{2}$ and $\cot(\pi/4) = 1$.

So, the lower limit value is $\sqrt{2} - 1$. Finally, we subtract:

$$I = (\sqrt{2} + 1) - (\sqrt{2} - 1) = \sqrt{2} + 1 - \sqrt{2} + 1 = 2$$

The result is a clean, elegant **2**. By breaking down the problem, using the right identities, and carefully managing our signs, we turned a daunting expression into a simple, satisfying answer.

Animated Solution for Mathematics - Definite Integration: $\int_{π /4}^{3 π /4} \frac{d x}{1 + c o s x}$ is equal to

Select Answer:

Visualized Solution

Visualizing the Integral

Rationalizing the Denominator

Applying Trigonometric Identity

Splitting the Fraction

Converting to Standard Integrals

Integrating the Terms

Applying the Upper Limit

Applying the Lower Limit

Final Calculation

The Sigma Insight: Fundamental Theorem & Properties of Definite Integrals

Analyzing the Setup

The Conjugate Key

The Art of Splitting

The Final Integration

The Evaluation

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