Class 12 Integrals · Short Answer2 Marksindefinite_integration
Class 12 Maths · Integrals · Short Answer #47
Evaluate \(\int \frac{\tan^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)}}\,dx\). Work on intervals with cos x≠0.
✅ Step-by-Step Solution
Final Answer: \(\frac{\sec^{5}{\left(x \right)}}{5} - \frac{\sec^{3}{\left(x \right)}}{3}\) + C
- Write tan³x sec³x=tan²x sec²x·sec x tan x. Set u=sec x, du=sec x tan x dx.
- Since tan²x=u²−1, integrate (u²−1)u²=u⁴−u² to u⁵/5−u³/3.
- Therefore the required integral is \(\frac{\sec^{5}{\left(x \right)}}{5} - \frac{\sec^{3}{\left(x \right)}}{3}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Write tan³x sec³x=tan²x sec²x·sec x tan x. Set u=sec x, du=sec x tan x dx.
- [1 mark] Since tan²x=u²−1, integrate (u²−1)u²=u⁴−u² to u⁵/5−u³/3.
- [0.5 marks] Therefore the required integral is \(\frac{\sec^{5}{\left(x \right)}}{5} - \frac{\sec^{3}{\left(x \right)}}{3}\) + C.
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