Evaluate the Integral

Problem

(∫_^)((x9)/(1+x20)*d(x))

Solution

1. Identify the substitution by recognizing that the numerator x9 is a multiple of the derivative of x10

2. Rewrite the denominator to express x20 as a power of x10

(∫_^)((x9)/(1+(x10)2)*d(x))

3. Substitute u=x10 which implies that the derivative is d(u)/d(x)=10*x9

4. Rearrange the differential to solve for the numerator's terms, giving x9*d(x)=1/10*d(u)

5. Substitute these values into the integral to change the variable from x to u

(∫_^)(1/(1+u2)⋅1/10*d(u))

6. Factor out the constant from the integral.

1/10*(∫_^)(1/(1+u2)*d(u))

7. Integrate using the standard integral formula (∫_^)(1/(1+u2)*d(u))=arctan(u)+C

1/10*arctan(u)+C

8. Back-substitute u=x10 to return to the original variable.

1/10*arctan(x10)+C

Final Answer

(∫_^)((x9)/(1+x20)*d(x))=1/10*arctan(x10)+C

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