Evaluate the Integral

Problem

(∫_^)(√(,t6+5*t)*(6*t5+5)*d(t))

Solution

1. Identify the inner function of the square root and its derivative to determine if usubstitution is appropriate.

2. Substitute u=t6+5*t

3. Differentiate u with respect to t to find d(u)=(6*t5+5)*d(t)

4. Rewrite the integral in terms of u

(∫_^)(√(,u)*d(u))

5. Apply the power rule for integration, (∫_^)(un*d(u))=(u(n+1))/(n+1)+C where n=1/2

(u(3/2))/(3/2)+C

6. Simplify the coefficient:

2/3*u(3/2)+C

7. Back-substitute the original expression for u

2/3*(t6+5*t)(3/2)+C

Final Answer

(∫_^)(√(,t6+5*t)*(6*t5+5)*d(t))=2/3*(t6+5*t)(3/2)+C

Want more problems? Check here!