   Chapter 7.P, Problem 2P

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# Evaluate ∫ 1 x 7 − x d x The straightforward approach would be to start with partial fractions, but that would be brutal. Try a substitution.

To determine

To evaluate:

The given integral by substitution method.

Explanation

Consider 1x7xdx

Hence, use the integration substitution method

Now according to the given question, the value of the integration of the given expression

1x7xdx is

1x7xdx=x2x3(x61)dx=3x23x3(x61)dx

By substitution

x3=t3x2dx=dt

3x23x3(x61)dx=13t(t21)dt=16(1t12t+1t+1)dt=161t1dt131tdt

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