If α = ∫₀¹ (e^(9x + 3 tan⁻¹ x)) ((12 + 9x²)/(1 + x²)) dx where tan⁻¹ x takes only principal values,

Q: Wait, why do we choose $u = 9x + 3 \tan^{-1} x$? How do I know that's the right substitution?

That is a fantastic question! In JEE-level integration, when you see an exponential function $e^{f(x)}$ multiplied by another function $g(x)$, it is almost always a hint that $g(x)$ is the derivative of $f(x)$. By choosing $u = f(x)$, you are essentially 'unlocking' the integral. Always look for that pattern first—it is the most common 'trap' or 'key' in these types of problems.

Q: Do I really need to change the limits of integration? Can't I just integrate and then substitute back?

You *could* substitute back, but changing the limits is much safer and faster. When you change the variable to $u$, the limits $x=0$ and $x=1$ no longer apply to $u$. Calculating the new limits immediately prevents the common mistake of plugging $x$ values into a $u$ expression. It keeps your work clean and organized, which is crucial under exam pressure.

Q: What if the derivative didn't match perfectly? What if there was a constant missing?

If the derivative was off by a constant factor (like a 2 or a 3), you would simply adjust the integral by multiplying and dividing by that constant. However, in this problem, the match is perfect. Always perform the differentiation of your chosen $u$ first; if it doesn't match, you know you need to rethink your substitution before you get lost in the algebra.

Question

If $\alpha = \int_{0}^{1} (e^{9x + 3 	an^{-1} x}) \left( \frac{12 + 9x^2}{1 + x^2} ight) dx$ where $	an^{-1} x$ takes only principal values, then the value of $(\log_e |1 + \alpha| - \frac{3\pi}{4})$ is

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a massive, locked gate. The gate is the integral $\alpha = \int_{0}^{1} (e^{9x + 3 	an^{-1} x}) \left( \frac{12 + 9x^2}{1 + x^2} ight) dx$.

It looks imposing, but in the world of JEE Advanced, we do not brute-force these problems; we look for the hidden symmetry.

### The Detective Work

Whenever you encounter an exponential function in an integral, your internal alarm should ring. The derivative of $e^{f(x)}$ is $e^{f(x)} \cdot f'(x)$.

If we can identify $f(x)$ and its derivative $f'(x)$ within the integrand, the problem collapses. Let $u = 9x + 3 	an^{-1} x$.

Now, let us perform the differentiation with the precision of a surgeon:

$$\frac{du}{dx} = \frac{d}{dx}(9x) + \frac{d}{dx}(3 	an^{-1} x) = 9 + \frac{3}{1 + x^2}$$

To see if this matches our multiplier, we combine these terms using a common denominator of $(1 + x^2)$:

$$\frac{du}{dx} = \frac{9(1 + x^2) + 3}{1 + x^2} = \frac{9 + 9x^2 + 3}{1 + x^2} = \frac{12 + 9x^2}{1 + x^2}$$

Look at that! The expression $\frac{12 + 9x^2}{1 + x^2}$ is exactly the rational function sitting in our integral. The gate is not just unlocked; it is wide open.

### The Transformation

When we change the variable from $x$ to $u$, we must also transform our boundaries. The integral is defined from $x=0$ to $x=1$.

For $x = 0$, we find $u = 9(0) + 3 	an^{-1}(0) = 0$.

For $x = 1$, we find $u = 9(1) + 3 	an^{-1}(1) = 9 + 3(\frac{\pi}{4}) = 9 + \frac{3\pi}{4}$.

With these new limits, our integral becomes a thing of beauty:

$$\alpha = \int_{0}^{9 + \frac{3\pi}{4}} e^u du$$

### The Final Reveal

Evaluating this is straightforward, as the integral of $e^u$ is simply $e^u$. Applying the Fundamental Theorem of Calculus, we get:

$$\alpha = [e^u]_0^{9 + \frac{3\pi}{4}} = e^{9 + \frac{3\pi}{4}} - e^0 = e^{9 + \frac{3\pi}{4}} - 1$$

We are almost at the finish line. The question asks for the value of $(\log_e |1 + \alpha| - \frac{3\pi}{4})$.

Rearranging our result for $\alpha$, we see that $1 + \alpha = e^{9 + \frac{3\pi}{4}}$. Taking the natural logarithm of both sides gives us:

$$\log_e |1 + \alpha| = 9 + \frac{3\pi}{4}$$

Finally, subtracting $\frac{3\pi}{4}$ from this value leaves us with the elegant result of **$9$**.

### Reflection

What did we learn today? We learned that complexity is often a mask for simplicity.

When you see a terrifying integral, do not panic. Look for the derivative, trust your substitution, and watch as the most daunting expressions simplify into something as clean as the number **$9$**.

Animated Solution for Mathematics - Definite Integration: If $α = \int_{0}^{1} (e^{9 x + 3 t a n^{- 1} x}) (\frac{12 + 9 x ^{2}}{1 + x ^{2}}) d x$ where $tan^{- 1} x$ takes only principal values, then the value of $(lo g_{e} ∣1 + α ∣ - \frac{3 π}{4})$ is

Enter Numerical Value:

Visualized Solution

Analyze the Integrand Structure

Define Substitution $u$

Calculate the Differential $d u$

Simplify the Differential

Confirm the Substitution

Transform the Lower Limit

Transform the Upper Limit

Evaluate the Simplified Integral

Rearrange the Equation

Final Calculation

The Sigma Insight: Integration by Substitution

Analyzing the Setup

The Detective Work

The Transformation

The Final Reveal

Reflection

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