Ex 7.6

Chapter 7 Class 12 Integrals
Serial order wise

### Transcript

Ex 7.6, 10 (sin^(β1)β‘π₯ )^2 β«1β(sin^(β1)β‘π₯ )^2 ππ₯ Let sin^(β1)β‘π₯=π β΄ π₯=sinβ‘π Differentiating both sides π€.π.π‘.π₯ ππ₯/ππ=cosβ‘π ππ₯=cosβ‘π.ππ Thus, our equation becomes β«1β(sin^(β1)β‘π₯ )^2 ππ₯ = β«1βπ^2 . cosβ‘π.ππ =π^2 β«1βγcosβ‘π.ππγββ«1β(π(π^2 )/ππ β«1βγcosβ‘π.ππγ)ππ =π^2 sinβ‘πββ«1βγ2π.sinβ‘π. ππγ =π^2 sinβ‘πβ2β«1βγπ½ πππβ‘π½. ππ½γ We know that β«1βγπ(π₯) πβ‘(π₯) γ ππ₯=π(π₯) β«1βπ(π₯) ππ₯ββ«1β(π^β² (π₯) β«1βπ(π₯) ππ₯) ππ₯ Putting f(x) = ΞΈ2 and g(x) = cos ΞΈ Solving I1 β«1βγπ½ πππβ‘π½. ππ½γ = πβ«1βγsinβ‘π ππγββ«1β(ππ/ππ β«1βγsinβ‘π ππγ) ππ = π(βcosβ‘π )ββ«1βγ1.(βcosβ‘π ) γ ππ = βπ cosβ‘π+β«1βcosβ‘π ππ = (βπ cosβ‘π+sinβ‘π )+πΆ1 Now we know that β«1βγπ(π₯) πβ‘(π₯) γ ππ₯=π(π₯) β«1βπ(π₯) ππ₯ββ«1β(πβ²(π₯)β«1βπ(π₯) ππ₯) ππ₯ Putting f(x) = ΞΈ and g(x) = sin ΞΈ Putting value of I1 in our equation β«1β(sin^(β1)β‘π₯ )^2 ππ₯ = π^2 sinβ‘πβ2β«1βγπ½ πππβ‘π½. ππ½γ =π^2 sinβ‘πβ2(βπ cosβ‘π+sinβ‘π+πΆ1) =π^2 sinβ‘π+2π cosβ‘πβ2 sinβ‘πβ2πΆ1 =π^2 sinβ‘π+2πβ(1βsin^2β‘γπ γ )β2 sinβ‘πβπͺπ =π^2 sinβ‘π+2πβ(1βsin^2β‘γπ γ )β2 sinβ‘π+πͺ =(γπ¬π’π§γ^(βπ)β‘π )^π π+π(γπ¬π’π§γ^(βπ)β‘π ) β(γπβγβ‘γπ^π γ )βππ+πͺ ππ πππ π=sin^(β1)β‘π₯ & sinβ‘π=π₯

#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.