Evaluate the Integral

Problem

(∫_^)(2*x2√(4,5+4*x3)*d(x))

Solution

1. Identify the substitution by looking for a function whose derivative is present in the integrand. Let u=5+4*x3

2. Differentiate u with respect to x to find d(u)

d(u)/d(x)=12*x2

d(u)=12*x2*d(x)

3. Solve for the term x2*d(x) to substitute it into the integral.

1/12*d(u)=x2*d(x)

4. Substitute u and d(u) into the original integral.

(∫_^)(2√(4,u)⋅1/12*d(u))

1/6*(∫_^)(u(1/4)*d(u))

5. Integrate using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

1/6⋅(u(5/4))/(5/4)+C

1/6⋅4/5*u(5/4)+C

2/15*u(5/4)+C

6. Back-substitute the original expression for u

2/15*(5+4*x3)(5/4)+C

Final Answer

(∫_^)(2*x2√(4,5+4*x3)*d(x))=2/15*(5+4*x3)(5/4)+C

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