Evaluate the Integral

Problem

(∫_1^2)((e^1/(x^5))/(x^6)*d(x))

Solution

1. Identify a suitable substitution to simplify the exponent of the natural exponential function.

2. Let u=1/(x^5) which can be rewritten as u=x^(−5)

3. Differentiate u with respect to x to find the differential d(u)

d(u)/d(x)=−5*x^(−6)

d(u)=−5/(x^6)*d(x)

4. Rearrange the differential to match the terms in the integrand.

−1/5*d(u)=1/(x^6)*d(x)

5. Determine the new limits of integration by substituting the original x values into the equation for u

Lower limit: *x=1⇒u=1/1^5=1

Upper limit: *x=2⇒u=1/2^5=1/32

6. Substitute the variables and limits into the integral.

(∫_1^1/32)(−1/5*e^u*d(u))

7. Apply the property of integrals to reverse the limits and remove the negative sign.

1/5*(∫_1/32^1)(e^u*d(u))

8. Integrate the function e^u

1/5*[e^u]^1_1/32

9. Evaluate the definite integral by subtracting the value at the lower limit from the value at the upper limit.

1/5*(e^1−e^1/32)

Final Answer

(∫_1^2)((e^1/(x^5))/(x^6)*d(x))=(e−√(32,e))/5

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