Example 611 substitution in a definite integral

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EXAMPLE 6.11Substitution in a Definite IntegralInvolving an ExponentialCompute150tet2/2dt.0
400CHAPTER 4..Integration4-58EXERCISES 4.6WRITING EXERCISES1.It is neverwrongto make a substitution in an integral, but some-times it is not very helpful. For example, using the substitutionu=x2, you can correctly conclude thatx3x2+1dx=12uu+1du,but the new integral is no easier than the original integral.In this case, a better substitution makes this workable. (Canyou find it?) However, the general problem remains of howyou can tell whether or not to give up on a substitution. Givesome guidelines for answering this question, using the integralsxsinx2dxandxsinx3dxas illustrative examples.In exercises 5–30, evaluate the indicated integral.5.x3x4+3dx6.sec2xtanx dx7.sinxcosxdx8.sin3xcosx dx9.x2cosx3dx10.sinx(cosx+3)3/4dx
3.Suppose that an integrand has a term of the formef(x). For ex-ample, suppose you are trying to evaluatex2ex3dx. Discusswhy you should immediately try the substitutionu=f(x). Ifthis substitution does not work, what could you try next? (Hint:4.Suppose that an integrand has a composite function of the form
f(g(x)). Explain why you should look to see if the integrandalso has the termg(x). Discuss possible substitutions.

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