Question:
The integral $\int \frac{1}{\cos^2 x (1 - \tan x)^2} \, dx =$
The integral $\int \frac{1}{\cos^2 x (1 - \tan x)^2} \, dx =$
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Always look for the derivative of a term present in the numerator. Here, $\sec^2 x$ is the derivative of $\tan x$, signaling a direct $u$-substitution.
Updated On: Jun 3, 2026
- $\frac{1}{1 - \tan x} + C$
- $-\frac{1}{1 - \tan x} + C$
- $\frac{1}{(1 - \tan x)^2} + C$
- $-\frac{1}{(1 - \tan x)^2} + C$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
We use integration by substitution. First, we rewrite the trigonometric integrand using the identity $\frac{1}{\cos^2 x} = \sec^2 x$.
Step 2: Meaning
Let $I = \int \frac{\sec^2 x}{(1 - \tan x)^2} \, dx$. Since $\sec^2 x$ is the derivative of $\tan x$, we substitute $u = 1 - \tan x$.
Step 3: Analysis
Let: \[ u = 1 - \tan x \implies du = -\sec^2 x \, dx \implies \sec^2 x \, dx = -du \] Substituting these into the integral: \[ I = \int \frac{-du}{u^2} = -\int u^{-2} \, du \] \[ I = -\left( \frac{u^{-1}}{-1} \right) + C = \frac{1}{u} + C \] Substituting $u = 1 - \tan x$ back: \[ I = \frac{1}{1 - \tan x} + C \]
Step 4: Conclusion
The value of the indefinite integral is $\frac{1}{1 - \tan x} + C$.
Final Answer: (A)
We use integration by substitution. First, we rewrite the trigonometric integrand using the identity $\frac{1}{\cos^2 x} = \sec^2 x$.
Step 2: Meaning
Let $I = \int \frac{\sec^2 x}{(1 - \tan x)^2} \, dx$. Since $\sec^2 x$ is the derivative of $\tan x$, we substitute $u = 1 - \tan x$.
Step 3: Analysis
Let: \[ u = 1 - \tan x \implies du = -\sec^2 x \, dx \implies \sec^2 x \, dx = -du \] Substituting these into the integral: \[ I = \int \frac{-du}{u^2} = -\int u^{-2} \, du \] \[ I = -\left( \frac{u^{-1}}{-1} \right) + C = \frac{1}{u} + C \] Substituting $u = 1 - \tan x$ back: \[ I = \frac{1}{1 - \tan x} + C \]
Step 4: Conclusion
The value of the indefinite integral is $\frac{1}{1 - \tan x} + C$.
Final Answer: (A)
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