Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #106
Evaluate \(\int_0^{\pi/4}(\sin x+\cos x)/(16+9\sin2x)dx\).
✅ Step-by-Step Solution
Final Answer: (ln2)/15
- Let u=sin x−cos x, du=(cos x+sin x)dx, with limits −1,0. Since u²=1−sin2x, the denominator is 25−9u².
- Use 1/(25−9u²)=(1/10)[1/(5−3u)+1/(5+3u)]. A primitive is (1/30)ln|(5+3u)/(5−3u)|.
- At u=0 the log is 0; at u=−1 the ratio is 2/8=1/4. Thus I=ln4/30=ln2/15.
- Therefore the integral equals (ln2)/15.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let u=sin x−cos x, du=(cos x+sin x)dx, with limits −1,0. Since u²=1−sin2x, the denominator is 25−9u².
- [1 mark] Use 1/(25−9u²)=(1/10)[1/(5−3u)+1/(5+3u)]. A primitive is (1/30)ln|(5+3u)/(5−3u)|.
- [1 mark] At u=0 the log is 0; at u=−1 the ratio is 2/8=1/4. Thus I=ln4/30=ln2/15.
- [0.5 marks] Therefore the integral equals (ln2)/15.
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