Definite Integration by properties - P4

Chapter 7 Class 12 Integrals
Concept wise

### Transcript

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