Definite Integration by properties - P4

Chapter 7 Class 12 Integrals
Concept wise

### Transcript

Ex 7.10, 4 By using the properties of definite integrals, evaluate the integrals : β«_0^(π/2)β(cos^5β‘π₯ ππ₯)/(sin^5β‘π₯ + cos^5β‘π₯ ) Let I=β«_0^(π/2)βγcos^5β‘π₯/(sin^5β‘π₯ + cos^5β‘π₯ ) ππ₯γ I= β«_0^(π/2)βγ(cos^5 (π/2 β π₯))/(γπ ππγ^5 (π/2 β π₯) + γπππ γ^5 (π/2 β π₯) ) ππ₯γ β΄ I = β«_0^(π/2)βγ sin^5β‘π₯/(cos^5β‘π₯ + sin^5β‘π₯ ) ππ₯γ Adding (1) and (2) i.e. (1) + (2) I+I=(γπππ γ^5 π₯)/(γπ ππγ^5 π₯ + γπππ γ^5 π₯) ππ₯+β«_0^(π/2)βγsin^5β‘π₯/(cos^5β‘π₯ + sin^5β‘π₯ ) ππ₯γ 2I=β«_0^(π/2)βγ[(γπππ γ^5 π₯ + γπ ππγ^5 π₯)/(γπππ γ^5 π₯ + γπ ππγ^5 π₯)] ππ₯γ 2I= β«_0^(π/2)βγ ππ₯γ I=1/2 β«_0^(π/2)βγ ππ₯γ I=1/2 [π₯]_0^(π/2) I=1/2 [π/2β0] π°=π/π