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=12u√u+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.sec2x√tanx dx7.sinx√cosxdx8.sin3xcosx dx9.x2cosx3dx10.sinx(cosx+3)3/4dx
Get answer to your question and much more
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
Get answer to your question and much more
f(g(x)). Explain why you should look to see if the integrandalso has the termg(x). Discuss possible substitutions.
Get answer to your question and much more