Evaluate the Integral

Problem

(∫_1^2)((e1/(x5))/(x6)*d(x))

Solution

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

2. Let u=1/(x5) 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/(x6)*d(x)

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

−1/5*d(u)=1/(x6)*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=1

Upper limit: *x=2⇒u=1/2=1/32

6. Substitute the variables and limits into the integral.

(∫_1^1/32)(−1/5*eu*d(u))

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

1/5*(∫_1/32^1)(eu*d(u))

8. Integrate the function eu

1/5*[eu]11/32

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

1/5*(e1−e1/32)

Final Answer

(∫_1^2)((e1/(x5))/(x6)*d(x))=(e−√(32,e))/5

Want more problems? Check here!