Evaluate the Integral

Problem

(∫_^)(1/(√(3,x)+√(4,x))*d(x))

Solution

1. Identify the least common multiple of the indices of the roots, which are 3 and 4 The least common multiple is 12

2. Substitute x=u12 to eliminate the radicals. This implies d(x)=12*u11*d(u)

3. Rewrite the integral in terms of u by substituting √(3,x)=u4 and √(4,x)=u3

(∫_^)((12*u11)/(u4+u3)*d(u))

4. Simplify the integrand by factoring u3 from the denominator and canceling it with the numerator.

(∫_^)((12*u8)/(u+1)*d(u))

5. Perform polynomial long division on (u8)/(u+1) to express the integrand as a polynomial plus a remainder.

u8=(u+1)*(u7−u6+u5−u4+u3−u2+u−1)+1

6. Integrate the resulting expression term by term.

12*(∫_^)((u7−u6+u5−u4+u3−u2+u−1+1/(u+1))*d(u))

7. Apply the power rule and the logarithmic rule for integration.

12*((u8)/8−(u7)/7+(u6)/6−(u5)/5+(u4)/4−(u3)/3+(u2)/2−u+ln(u+1))+C

8. Back-substitute u=x(1/12) to return to the original variable x

(3*x(2/3))/2−(12*x(7/12))/7+2*x(1/2)−(12*x(5/12))/5+3*x(1/3)−4*x(1/4)+6*x(1/6)−12*x(1/12)+12*ln(x(1/12)+1)+C

Final Answer

(∫_^)(1/(√(3,x)+√(4,x))*d(x))=(3*x(2/3))/2−(12*x(7/12))/7+2*x(1/2)−(12*x(5/12))/5+3*x(1/3)−4*x(1/4)+6*x(1/6)−12*x(1/12)+12*ln(√(12,x)+1)+C

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