Evaluate the Integral integral of 3(2t+5)^3 with respect to t

Problem

(∫_^)(3*(2*t+5)3*d(t))

Solution

1. Identify the integral as a power of a linear function, which can be solved using usubstitution.

2. Substitute u=2*t+5

3. Differentiate u with respect to t to find d(u)=2*d(t) which implies d(t)=1/2*d(u)

4. Rewrite the integral in terms of u by substituting the expressions for 2*t+5 and d(t)

(∫_^)(3*u3⋅1/2*d(u))

5. Factor out the constants from the integral.

3/2*(∫_^)(u3*d(u))

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

3/2⋅(u4)/4+C

7. Simplify the resulting expression.

(3*u4)/8+C

8. Back-substitute u=2*t+5 to return to the original variable t

(3*(2*t+5)4)/8+C

Final Answer

(∫_^)(3*(2*t+5)3*d(t))=(3*(2*t+5)4)/8+C

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