Evaluate ∫x2sin2xdx. Illustrate and check whether your answer is reasonable by gaphing both the function and its antiderivative suppose that c=0.
If we use 2=2x, then x=z2 so dx=12dz
∫x2sin2xdx=∫(z2)2(sinz)(dz2)=18∫z2sinzdz
By using integration by parts,
If we let u=z2 and dv=sinzdz. Then,
du=2zdz and v=−cosz
Thus,
18∫z2sinzdz=uv−∫vdu=−z2cosz−∫(−cosz)(2zdz)=−z2cosz+2∫zcoszdz
Again by using integration by parts, if we let u1=z and dv1=coszdz, then
du1=dz and v1=sinz
so, ∫zcoszdz=u1v1−∫vzdu1=zsinz−∫sinzdz=zsinz−(−cosz)=zsinz+cosz
Going back to the first equation,
18∫z2sinzdz=18[−z2cosz+2(zsinz+cosz)]=−z2cosz8+zsinz4+cosz4+c
but z=2x, therefore,
∫x2sin2xdx=−(2x)28cos(2x)+(2x)sin(2x)4+cos2x4+c=−x2cos(2x)2+x2sin(2x)+cos2x4+c
We can see from the graph that our answer is reasonable, because the graph of the anti-derivative f is increasing when f′ is positive. On the other hand, the graph of f is decreasing when f′ is negative.
Friday, August 17, 2012
Single Variable Calculus, Chapter 8, 8.1, Section 8.1, Problem 42
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