Ex 7.3

Chapter 7 Class 12 Integrals
Serial order wise

### Transcript

Ex 7.3, 9 Integrate (πππ  π₯)/(1 + πππ  π₯) β«1βγ(πππ  π₯)/(1 + πππ  π₯) " " ππ₯γ = β«1β((cosβ‘π₯ + 1 β 1)/(1 + cosβ‘π₯ )) ππ₯ =β«1β((1 + cosβ‘π₯ β 1)/(1 + cosβ‘π₯ )) ππ₯ =β«1β((1 + cosβ‘π₯)/(1 + cosβ‘π₯ ) β 1/(1 + cosβ‘π₯ )) ππ₯ =β«1βγ1β1/(1 + cosβ‘π₯ )γ ππ₯ =β«1β1 ππ₯ββ«1βπ/(π + πππβ‘π ) ππ₯ =β«1β1 ππ₯ββ«1β1/(π γπππγ^πβ‘γπ/πγ ) ππ₯ =β«1β1 ππ₯ββ«1β1/2 sec^2β‘γπ₯/2γ ππ₯ =β«1β1 ππ₯β1/2 β«1βsec^2β‘γπ₯/2γ ππ₯ =π₯β 1/2 γtan γβ‘γπ₯/2γ/(1/2) +πΆ =π₯β 2/2 γtan γβ‘γπ₯/2γ +πΆ =πβ γπ­ππ§ γβ‘γπ/πγ +πͺ β«1βsec^2β‘(ππ₯+π) ππ₯=π‘ππβ‘(ππ₯ + π)/π +πΆ We know that cos 2π=2 cos^2β‘γπβ1γ cosβ‘2π+1=2 cos^2β‘π Replacing π by π₯/2 cosβ‘2(π₯/2)+1=2 cos^2β‘γπ₯/2γ cosβ‘π₯+1=2 cos^2β‘γπ₯/2γ