Evaluate the Integral

Problem

(∫_0^1)(x/((x2+1)3)*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand. Let u=x2+1

2. Differentiate the substitution to find the relationship between d(u) and d(x)

d(u)/d(x)=2*x

d(u)=2*x*d(x)

1/2*d(u)=x*d(x)

3. Change the limits of integration to correspond with the variable u

Lower limit: *x=0⇒u=0+1=1

Upper limit: *x=1⇒u=1+1=2

4. Substitute the expressions for u d(u) and the new limits into the integral.

(∫_1^2)(1/(2*u3)*d(u))

5. Rewrite the integrand using a negative exponent to prepare for integration.

1/2*(∫_1^2)(u(−3)*d(u))

6. Apply the power rule for integration, which states (∫_^)(un*d(u))=(u(n+1))/(n+1)

1/2*[(u(−2))/(−2)]21

[−1/(4*u2)]21

7. Evaluate the expression at the upper and lower limits.

(−1/(4*(2)2))−(−1/(4*(1)2))

−1/16+1/4

8. Simplify the final numerical result.

−1/16+4/16=3/16

Final Answer

(∫_0^1)(x/((x2+1)3)*d(x))=3/16

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