JEE Main Integration Shortcut — Definite Integrals Made Easy 💡

 

❓ Concept

Integration Trick – tan⁻¹ Form in 60 Sec!

Whenever you see an integral of the type

$$\int \frac{\sin x}{a+b{\cos }^{2}x}dx$$

(or with extra terms in the numerator), understand one thing clearly 👇
👉 This is a pure tan⁻¹ game.

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✍️ Short Explanation

This type of integral is very common in JEE Main + Advanced.
The key idea is:

• sin x dx is the derivative of cos x

• The denominator becomes a quadratic in cos x

• Final answer always involves tan⁻¹

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1️⃣ Step 1 — Identify the Pattern

Integral of the form:

$$\int \frac{\text{(something)}\cdot \sin x}{a+b{\cos }^{2}x}dx$$

👉 Always try substitution:

$$u = \cos x$$
$$du = -\sin x\,dx$$

This instantly simplifies the integral.

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2️⃣ Step 2 — Apply Substitution

From substitution:

$$\sin x\,dx = -du$$

For definite integrals, limits change:

$$x = 0 \Rightarrow u = \cos 0 = 1$$
$$x = \pi \Rightarrow u = \cos \pi = -1$$

So the integral converts into:

$$\int \frac{-(\text{numerator})\,du}{1 + 3u^2}$$

👉 Now it becomes a rational function in u, which is easy.

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3️⃣ Step 3 — Split the Integral (Most Important JEE Step)

If the numerator looks like:

$$(x+3)\sin x$$

Split it:

$$(x+3) = x + 3$$

So the integral becomes:

$$\int \frac{x\sin x}{1+3\cos^2 x}\,dx \;+\; \int \frac{3\sin x}{1+3\cos^2 x}\,dx$$

✔ First part may simplify or vanish (especially in symmetric limits)
✔ Second part is direct tan⁻¹ form

This splitting trick saves huge time in exams.

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4️⃣ Step 4 — Use the Standard Formula

Remember this golden formula:

$$\int \frac{du}{a + bu^2} = \frac{1}{\sqrt{ab}}\tan^{-1}\!\left(u\sqrt{\frac{b}{a}}\right)$$

For this case:

$$a = 1,\quad b = 3$$

So,

$$\int \frac{du}{1 + 3u^2} = \frac{1}{\sqrt{3}}\tan^{-1}(u\sqrt{3})$$

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5️⃣ Final JEE Concept Summary

🔥 Master Rule for Such Integrals:

• sinx present? → Put u = cosx

• Denominator quadratic in cosx → tan⁻¹ guaranteed

• Split numerator smartly

• Use standard formula

• Apply limits carefully

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✅ Final Takeaway

👉 sin x / (a + b cos² x)
→ Always substitute u = cos x
→ Convert to tan⁻¹ form
→ Solve in under 60 seconds

This one trick alone cracks multiple JEE PYQs.