ExampleEvaluate the indefinite integralZ6x3√5x4+ 12dx.Solution:This problem is made difficult because of the square root of afunctionofxin thedenominator.However, a simplification occurs when we make an appropriate substitution.Letu=u(x) denote the function ofxunder the square root, i.e.,u= 5x4+ 12,(1)and note that the differentialdusatisfiesdu= 20x3dx.(2)The reason thatuis a good substitution in this problem is that, after replacing the expression√5x4+ 12 in the integrand by√u, the remainingx-dependent factor,x3dx, can be seen tobe a constant multiple ofdugiven above.That is, we can divide through by 20 in (2) toobtainx3dx=120du,(3)and obtain a simplification of the original integral occurs when the substitutions (1) and (3)
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