Evaluate the Integral

Problem

(∫_^)(√(,t6+7*t)*(6*t5+7)*d(t))

Solution

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

2. Let u be the expression inside the square root.

u=t6+7*t

3. Differentiate u with respect to t to find d(u)

d(u)/d(t)=6*t5+7

d(u)=(6*t5+7)*d(t)

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

(∫_^)(√(,u)*d(u))

5. Rewrite the square root as a fractional exponent to prepare for integration.

(∫_^)(u(1/2)*d(u))

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

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

7. Simplify the coefficient by multiplying by the reciprocal.

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

8. Substitute the original expression for u back into the result.

2/3*(t6+7*t)(3/2)+C

Final Answer

(∫_^)(√(,t6+7*t)*(6*t5+7)*d(t))=2/3*(t6+7*t)(3/2)+C

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