Evaluate the indefinite integral. (Use C for the constant of integration.) ∫ecos⁡(9t)sin⁡(9t)dt

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Evaluate the indefinite integral. (Use C for the constant of integration.)  ∫ecos⁡(9t)sin⁡(9t)dt

Howell · Accepted Answer

Step 1  Given integral:  ∫ecos⁡(9t)sin⁡(9t)dt  Step 2  Now,  ∫ecos⁡(9t)sin⁡(9t)dt  Sustitute:  cos⁡(9t)=u  Differentiate both sides:  ⇒−sin⁡(9t)×9dt=du   [∵d(cos⁡(x))dx=−sin⁡(x)]  ⇒sin⁡(9t)dt=−du9  Now the integral becomes:  ∫−19eudu=−19∫eudu   [∵∫exdx=ex+c]  =−19eu+C  Substitute back the value of u:  =−19ecos⁡(9t)+C

Roger Noah · Accepted Answer

Step 1: Use Integration by Substitution.  Let u=cos⁡9t,du=−9sin⁡9tdt, then sin⁡9tdt=−19du  Step 2: Using u and du above, rewrite ∫ecos⁡9tsin⁡9tdt.  ∫−eu9du  Step 3: Use Constant Factor Rule: ∫cf(x)dx=c∫f(x)dx.  −19∫eudu  Step 4: The integral of ex is ex.  −eu9  Step 5: Substitute u=cos⁡9t back into the original integral.  −ecos⁡9t9  Step 6: Add constant.  −ecos⁡9t9+C

Evaluate the indefinite integral. (Use C for the constant of

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