Evaluate the following integral. etan⁡xcos2xdx

Question

Evaluate the following integral.  etan⁡xcos2xdx

movingsupplyw1 · Accepted Answer

Step 1 Given integral is etan⁡xcos2xdx We know that sec⁡x=1cos⁡x Therefore, given integral can be written as ∫etan⁡xcos2xdx=∫etan⁡x(sec2xdx) Step 2 Now, we will use the following substitution to evaluate the integral tan⁡x=t sec2xdx=dt Therefore, we have ∫etan⁡xcos2xdx=∫etdt ∫etan⁡xcos2xdx=et+c ∫etan⁡xcos2xdx=etan⁡x+c Step 3 Ans: ∫etan⁡xcos2xdx=etan⁡x+c

Annie Gonzalez · Accepted Answer

Given:  ∫etan⁡(x)cos⁡(x)2dx  ∫1dt  Use ∫1dx=x to evaluate the integral  t  Substitute back  etan⁡(x)  Add C  Answer:  etan⁡(x)+C

Vasquez · Accepted Answer

∫etan⁡(x)cos⁡(x)2dx We put the expression 1/cos⁡(x)2 under the differential sign, i.e.: 1cos⁡(x)2dx=d(tan⁡(x)),t=tan⁡(x) Then the original integral can be written as follows: ∫etdt This is a tabular integral: ∫etdt=et+C To write down the final answer, it remains to substitute tan⁡(x) instead of t. etan⁡(x)+C

Evaluate the following integral. \frac{e^{\tan x}}{\cos^{2}x}dx

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