Question
Download Solution PDFWhat is \(\smallint {\sin ^3}x\cos x\;dx\) equal to?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
- \(\smallint {{\rm{x}}^{\rm{n}}}{\rm{dx}} = {\rm{\;}}\frac{{{{\rm{x}}^{\rm{n}+1}}}}{{\rm{n+1}}} + {\rm{c}}\)
Calculation:
Let I = \(\smallint {\sin ^3}x\cos x\;dx\)
Let sin x = t
Now differentiating both sides, we get
⇒ cos x dx = dt
Now,
\({\rm{I\;}} = \smallint {\sin ^3}{\rm{x}}\cos {\rm{x\;dx}} = {\rm{\;}}\smallint {{\rm{t}}^3}{\rm{dt}} = {\rm{\;}}\frac{{{{\rm{t}}^4}}}{4} + {\rm{c}}\)
\(\Rightarrow {\rm{I\;}} = {\rm{\;}}\frac{{{{\sin }^4}{\rm{x}}}}{4} + {\rm{c}} = {\rm{\;}}\frac{{{{\left( {1 - {{\cos }^2}{\rm{x}}} \right)}^2}}}{4} + {\rm{c\;}}\)
∴ Option 4 is correct answer
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