Miscellaneous

Chapter 7 Class 12 Integrals
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### Transcript

Misc 10 Integrate the function (γsin^8 π₯γβ‘β cos^8β‘π₯)/(1 β 2 sin^2β‘γπ₯ cos^2β‘π₯ γ ) β«1β(γsin^8 π₯γβ‘β cos^8β‘π₯)/(1 β 2 sin^2β‘γπ₯ cos^2β‘π₯ γ ) =β«1β((sin^4 π₯)^2β‘γβ γ (cos^4 π₯)^2)/(1 β 2 sin^2β‘γπ₯ cos^2β‘π₯ γ ) =β«1β((sin^4 π₯ + cos^4β‘π₯ )β‘(sin^4β‘π₯ β cos^4β‘π₯ )ππ₯)/(1 β 2 sin^2β‘γπ₯ cos^2β‘π₯ γ ) =β«1β((sin^4 π₯ + cos^4β‘π₯ )β‘γ ((sin^2 π₯)^2 β (cos^2 π₯)^2 )γ ππ₯)/(1 β 2 sin^2β‘γπ₯ cos^2β‘π₯ γ ) =β«1β((sin^4 π₯ + cos^4β‘π₯ )β‘(sin^2β‘π₯ + cos^2β‘π₯ ) (sin^2β‘π₯ β cos^2β‘π₯ )ππ₯)/(1 β 2 sin^2β‘γπ₯ cos^2β‘π₯ γ ) =β«1β((sin^4 π₯ + cos^4β‘π₯ )β‘(sin^2β‘π₯ + cos^2β‘π₯ ) (sin^2β‘π₯ β cos^2β‘π₯ )ππ₯)/(1 β 2 sin^2β‘γπ₯ cos^2β‘π₯ γ ) =β«1β(γ(sin^4 π₯ + cos^4β‘π₯ ) (1)γβ‘γ (sin^2β‘π₯ β cos^2β‘π₯ )γ ππ₯)/(1 β 2 sin^2β‘γπ₯ cos^2β‘π₯ γ ) Adding & Subtracting 2 sin^2β‘π₯ cos^2β‘π₯ =β«1β((sin^4 π₯ + cos^4β‘π₯ + 2 sin^2β‘π₯ cos^2β‘π₯ β 2 sin^2β‘cos^2β‘π₯ )β‘γ (sin^2β‘π₯ β cos^2β‘π₯ )γ ππ₯)/(1 β 2 sin^2β‘γπ₯ cos^2β‘π₯ γ ) =β«1β((((sin^2β‘π₯ )^2+ (cos^2β‘π₯ )^2 + 2 sin^2β‘π₯ cos^2β‘π₯ )β2 sin^2β‘π₯ cos^2β‘π₯ )β‘(sin^2β‘π₯ β cos^2β‘π₯ )ππ₯)/(1 β 2 sin^2β‘γπ₯ cos^2β‘π₯ γ ) =β«1β(γ((sin^2β‘π₯ + cos^2β‘π₯ )^2 β 2 sin^2β‘π₯ cos^2β‘π₯ ) γβ‘(sin^2β‘π₯ β cos^2β‘π₯ )ππ₯)/(1 β 2 sin^2β‘γπ₯ cos^2β‘π₯ γ ) =β«1β(γ(1^2 β 2 sin^2β‘π₯ cos^2β‘π₯ ) γβ‘(sin^2β‘π₯ β cos^2β‘π₯ )ππ₯)/(1 β 2 sin^2β‘γπ₯ cos^2β‘π₯ γ ) =β«1β(γ(1 β 2 sin^2β‘π₯ cos^2β‘π₯ ) γβ‘(sin^2β‘π₯ β cos^2β‘π₯ )ππ₯)/(1 β 2 sin^2β‘γπ₯ cos^2β‘π₯ γ ) =β«1β(sin^2β‘π₯βcos^2β‘π₯ ) ππ₯ =ββ«1β(cos^2β‘π₯βsin^2β‘π₯ ) ππ₯ =ββ«1βcosβ‘2π₯ . ππ₯ =(βπ)/π π¬π’π§β‘ππ+πͺ (sin^2β‘π₯ + cos^2β‘π₯=1" " ) (Using cos 2π=γπππ γ^2 πβγπ ππγ^2 π)