Example 21 - Chapter 7 Class 12 Integrals
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Example 21 Find β«1βπ^π₯ sinβ‘π₯ ππ₯ Let I1 = β«1βγ π^π₯ γ sinβ‘π₯ ππ₯ I1 = sinβ‘π₯ β«1βγπ^π₯ ππ₯γββ«1β(π(sinβ‘π₯ )/ππ₯ β«1βγπ^π₯ ππ₯γ) ππ₯ I1 = π^π₯ sinβ‘π₯ββ«1βγcosβ‘π₯ . π^π₯ ππ₯γ Now we know that β«1βγπ(π₯) πβ‘(π₯) γ ππ₯=π(π₯) β«1βπ(π₯) ππ₯ββ«1β(πβ²(π₯)β«1βπ(π₯) ππ₯) ππ₯ Putting f(x) = sin x and g(x) = ex Solving I2 I2 = β«1βγcosβ‘π₯ . π^π₯ ππ₯γ I2 = cos x β«1βγπ^π₯ ππ₯γ β β«1βγ((cosβ‘π₯)β²γ β«1βγπ^π₯ ππ₯γ)ππ₯ I2 = cos x π^π₯ β β«1βγ(βsinβ‘π₯)γ π^π₯ ππ₯ I2 = π^π₯ cos x + β«1βsinβ‘π₯ π^π₯ ππ₯ I2 = π^π₯ cos x + πΌ1 Now we know that β«1βγπ(π₯) πβ‘(π₯) γ ππ₯=π(π₯) β«1βπ(π₯) ππ₯ββ«1β(πβ²(π₯)β«1βπ(π₯) ππ₯) ππ₯ Putting f(x) = sin x and g(x) = ex Now, Putting value of I2 in (1) , I1 = " " π^π₯ sinβ‘π₯ββ«1βγcosβ‘π₯ π^π₯ γ ππ₯ I1 = " " π^π₯ sinβ‘π₯β(π^π₯ cosβ‘π₯+πΌ1)+πΆ I1 = " " π^π₯ sinβ‘π₯βπ^π₯ cosβ‘π₯βπΌ1+πΆ 2I1 = " " π^π₯ sinβ‘π₯βπ^π₯ cosβ‘π₯ + πΆ I1 = 1/2 (π^π₯ sinβ‘π₯βπ^π₯ cosβ‘π₯ ) + C π°π = π^π/π (πππβ‘πβπππβ‘π ) + C
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About the Author
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 and Computer Science at Teachoo