Question:
\( \int \frac{\cos x}{\sin^2 x (\sin x + \cos x)} \, dx \) is equal to
\( \int \frac{\cos x}{\sin^2 x (\sin x + \cos x)} \, dx \) is equal to
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In trig integrals, try substitution $\tan x$ or split expression smartly.
Updated On: Apr 23, 2026
- $\log\left|\frac{1+\tan x}{\tan x}\right| - \cot x + C$
- $\log\left|\frac{\tan x}{1+\tan x}\right| + C$
- $\log\left|\frac{\tan x}{1+\tan x}\right| - \tan x + C$
- $\log\left|\frac{\tan x}{1+\tan x}\right| + \cot x + C$
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The Correct Option is A
Solution and Explanation
Concept:
Split expression and use substitution with trig identities.
Step 1: Rewrite expression.
\[ \frac{\cos x}{\sin^2 x(\sin x + \cos x)} \]
Step 2: Use substitution $t = \tan x$.
\[ dt = \sec^2 x\, dx \]
Step 3: Convert terms.
\[ \sin x = \frac{t}{\sqrt{1+t^2}}, \quad \cos x = \frac{1}{\sqrt{1+t^2}} \]
Step 4: Simplify integral.
Expression reduces to sum of standard forms.
Step 5: Integrate.
\[ = \log\left|\frac{1+\tan x}{\tan x}\right| - \cot x + C \] Conclusion:
Answer = Option (A)
Step 1: Rewrite expression.
\[ \frac{\cos x}{\sin^2 x(\sin x + \cos x)} \]
Step 2: Use substitution $t = \tan x$.
\[ dt = \sec^2 x\, dx \]
Step 3: Convert terms.
\[ \sin x = \frac{t}{\sqrt{1+t^2}}, \quad \cos x = \frac{1}{\sqrt{1+t^2}} \]
Step 4: Simplify integral.
Expression reduces to sum of standard forms.
Step 5: Integrate.
\[ = \log\left|\frac{1+\tan x}{\tan x}\right| - \cot x + C \] Conclusion:
Answer = Option (A)
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