Question:
What is \(\int \sin^3 x \cos x\,dx\) equal to?
\(\int \sin^3 x \cos x\,dx\) किसके बराबर है?
Options:
- \(\frac{\cos^4 x}{4} + c\)
- \(\frac{\sin^4 x}{4} + c\)
- \(\frac{(1 – \sin^2 x)^2}{4} + c\)
- \(\frac{(1 – \cos^2 x)^2}{4} + c\)
- \(\frac{\cos^4 x}{4} + c\)
- \(\frac{\sin^4 x}{4} + c\)
- \(\frac{(1 – \sin^2 x)^2}{4} + c\)
- \(\frac{(1 – \cos^2 x)^2}{4} + c\)
Correct Answer: d)
Explanation:
Let \(u = \sin x\), then \(du = \cos x\,dx\).
\[\int u^3\,du = \frac{u^4}{4} + c = \frac{\sin^4 x}{4} + c\]
Since \(\sin^2 x = 1 – \cos^2 x\):
\[\frac{\sin^4 x}{4} + c = \frac{(\sin^2 x)^2}{4} + c = \frac{(1 – \cos^2 x)^2}{4} + c\]
According to the official answer sheet, option (d) is designated as the answer key.
Hence, the correct option is (d).
माना \(u = \sin x\), तो \(du = \cos x\,dx\):
\[\int u^3\,du = \frac{\sin^4 x}{4} + c\]
चूँकि \(\sin^2 x = 1 – \cos^2 x\) होता है, अतः इसे \(\frac{(1 – \cos^2 x)^2}{4} + c\) भी लिखा जा सकता है।
आधिकारिक उत्तर कुंजी के अनुसार सही विकल्प (d) है।