Find integral. ∫excsc⁡excot⁡exdx

Question

Find integral.  ∫excsc⁡excot⁡exdx

Karen Robbins · Accepted Answer

Step 1 To find: The integral of ∫excsc⁡excot⁡exdx. Formula used: Power rule of integration: ∫xndx=xn+1n+1+C Step 2 Calculation: Substitute ex=tand exdx=dt∈∫ex⋅csc⁡ex⋅cot⁡exdx. ∫csc⁡t⋅cot⁡tdt=∫1sin⁡t⋅cos⁡tsin⁡tdt =∫cos⁡tsin2tdt Substitute sin⁡t=u and cos⁡tdt=du in above integral: ∫cos⁡tsin2tdt=∫1u2du =∫u−2du =u−2+1−2+1+C =−1u+C Substitute sin⁡t=u in above integral: ∫cos⁡tsin2tdt=−1sin⁡t+C Substitute ex=t in above integral: ∫cos⁡tsin2tdt=−1sin⁡ex+C =−csc⁡ex+C Thus, the value of integral ∫ex⋅csc⁡ex⋅cot⁡exdx is −csc⁡ex+C.

Navreaiw · Accepted Answer

excot⁡(ex)csc⁡(ex)dx  =∫cot⁡(u)csc⁡(u)du  =−csc⁡(u)  =−csc⁡(ex)  ∫excot⁡(ex)csc⁡(ex)dx  =−csc⁡(ex)+C

nick1337 · Accepted Answer

∫ex×csc⁡(ex)cot⁡(ex)dx Transform the expression ∫ex×1sin⁡(ex)×cos⁡(ex)sin⁡(ex)dx Calculate ∫ex×cos⁡(ex)sin⁡(ex)2dx Transform the expression ∫1t2dt Evaluate the integral −1t Substitute back −1sin⁡(ex) Add C Solution −1sin⁡(ex)+C

Find integral. \int e^{x}\csc e^{x}\cot e^{x}dx

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