Evaluate the indefinite integral 5 / (t + 2)8 dt by making a u-substitution. (Gi

Question

Evaluate the indefinite integral 5 / (t + 2)8 dt by making a u-substitution. (Give a function of t here.) When you make the substitution, the integral becomes k g(u)du where g(u) = (Give a function of u here.) (Give the constant that would be needed outside the integral.) Now do the antidifferentiation. (Don't forget the +C. "C" must be capital not lowercase.) Give the final result by substituting the function of t back in. Evaluate the definite integral e sin(x) cos(x)dx by making a u-substitution. u = sinx (Give a function of x here.) du = cosxdx When you make the substitution, the integral becomes k g(u)du where g(u) = (Give a function of u here.) k = 1 (Give the constant that would be needed outside the integral.) Now do the antidifferentiation. (Omit the + C because it's a definite integral.) Give the final result by substituting the function of x back in then evaluating at the limits

Explanation / Answer

1) let us substitute u = t+2

then du =dt

integral will be 5/u^3 = -5/2u^2 = -5/2(t+2)^2 + c

When we make the substitution g(u) = 1/u^3 and k = 5

integral will be 5/u^3 = -5/2u^2 = -5/2(t+2)^2 + c

2) u = sinx

du = cosx dx

g(u) = e^(u) and k = 1

Integral will be 0 to 1 (e^(u) du ) = e^(1) - 1 = e - 1

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