Question

If I=∫\(\frac{x^2dx}{(x\,sin\,x+cos\,x)^2}\)=f(x)+tan x+c, then f(x) is

• A. \(\frac{sin\,x}{xsin\,x+cos\,x}\)
• B. \(\frac{1}{(xsin\,x+cos\,x)^2}\)
• C. \(\frac{-x}{cos\,x(xsin\,x+cos\,x)}\)
• D. \(\frac{1}{sin\,x(xcos\,x+sin\,x)}\)

Answer 1

The Correct Option is C

Step 1: Simplify the Integral

We are given:

\(I = \int \frac{x^2 \, dx}{(x \sin x + \cos x)^2}\)

Let's try to rewrite the integrand in a form that is easier to integrate. Divide both numerator and denominator by \(x^2\cos^2x\):

\(I = \int \frac{x^2 dx}{(x \sin x + \cos x)^2} = \int \frac{x^2 dx}{x^2\cos^2 x (\frac{x \sin x}{\cos x} + 1)^2/x^2} =\int \frac{sec^2(x)}{ (tan(x)+1/x)^2} dx \)

This doesn't simplify it.

Let \(u = x \sin x + \cos x\)

\(\frac{du}{dx} = x \cos x + \sin x - \sin x = x \cos x\)

So, let's try integration by parts. Rewrite as:

\(I = \int \frac{x}{\cos x} \cdot \frac{x \cos x \, dx}{(x \sin x + \cos x)^2}\)

Now, we will perform integration by parts:

\(\int u dv = uv - \int v du\)

Let \(u = \frac{x}{\cos x}\) and \(dv = \frac{x \cos x}{(x \sin x + \cos x)^2} \, dx\)

First, find du:

\(du = \frac{\cos x - x(-\sin x)}{\cos^2 x} dx = \frac{\cos x + x \sin x}{\cos^2 x} dx\)

Now, find v. Since \(dv = \frac{x \cos x}{(x \sin x + \cos x)^2} \, dx\), we have:

Let \(w = x \sin x + \cos x\) then \(dw = (x \cos x + \sin x - \sin x)dx = x cos x dx\)

\(v = \int \frac{x \cos x}{(x \sin x + \cos x)^2} \, dx = \int \frac{dw}{w^2} = -\frac{1}{w} = -\frac{1}{x \sin x + \cos x}\)

Step 2: Apply Integration by Parts

Using integration by parts, we have:

\(I = \frac{x}{\cos x} \cdot \left(-\frac{1}{x \sin x + \cos x}\right) - \int \left(-\frac{1}{x \sin x + \cos x}\right) \cdot \frac{\cos x + x \sin x}{\cos^2 x} \, dx\)

\(I = -\frac{x}{\cos x (x \sin x + \cos x)} + \int \frac{1}{\cos^2 x} \, dx\)

\(I = -\frac{x}{\cos x (x \sin x + \cos x)} + \int \sec^2 x \, dx\)

Step 3: Evaluate the Remaining Integral

We know that \(\int \sec^2 x \, dx = \tan x + C\)

So, we have:

\(I = -\frac{x}{\cos x (x \sin x + \cos x)} + \tan x + C\)

Step 4: Identify f(x)

We are given that \(I = f(x) + \tan x + C\). Comparing this with our result:

\(f(x) = -\frac{x}{\cos x (x \sin x + \cos x)}\)

Conclusion:

The function f(x) is:

\(f(x) = -\frac{x}{\cos x (x \sin x + \cos x)}\)