Find the general indefinite integrals. Integral of πsin⁡xcos3xdx

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Find the general indefinite integrals.  Integral of πsin⁡xcos3xdx

Richard Cheatham · Accepted Answer

Step 1  Consider the given function as πsin⁡xcos3xdx.  We have to find the general indefinite integral of πsin⁡xcos3xdx which is ∫πsin⁡xcos3xdx.  Take t=cos⁡x.  Then, cos3x=t3 and dt=−sin⁡dx.  Step 2  Substitute cos3x=t3 and dt=−sin⁡dx ∈ ∫πsin⁡xcos3xdx to evaluate the indefinite integral as follows.  ∫πsin⁡xcos3xdx=∫−π(cos3x)(−sin⁡xdx)  =∫−πt3dt  =−π∫t3dt  =−π(t44)  =−π(cos4x4)+C   [∵t=cos⁡x]  Therefore, the general indefinite integral of ∫πsin⁡xcos3xdx=−π4cos4x+C where C is the constant of integration.

Sharolyn Larson · Accepted Answer

Step 1: Regroup terms.  ∫πcos3xsin⁡xdx  Step 2: Use Constant Factor Rule: ∫cf(x)dx=c∫f(x)dx.  πcos3∫xsin⁡xdx  Step 3: Use Integration by Parts on ∫xsin⁡xdx.  Let u=x,dv=sin⁡x,du=dx,v=−cos⁡x  Step 4: Substitute the above into uv−∫vdu.  πcos3(−xcos⁡x−∫−cos⁡xdx)  Step 5: Use Trigonometric Integration: the integral of cos⁡x is sin⁡x.  πcos3(−xcos⁡x+sin⁡x)  Step 6: Add constant.  πcos3(−xcos⁡x+sin⁡x)+C

Find the general indefinite integrals. Integral of \pi \sin x \cos^{3}x

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