91Ó°ÊÓ

Finding \(u\) and \(d u\) In Exercises \(1-4,\) complete the table by identifying \(u\) and \(d u\) for the integral. $$ \begin{array}{l}{\int f(g(x)) g^{\prime}(x) d x \quad u=g(x) \quad d u=g^{\prime}(x) d x} \\ {\int \frac{\cos x}{\sin ^{2} x} d x}\end{array} $$

Evaluating a Definite Integral as a Limit In Exercises \(3-8\) , evaluate the definite integral by the limit definition. $$ \int_{-2}^{3} x d x $$

In Exercises 3–6, find the general solution of the differential equation and check the result by differentiation. $$ \frac{d y}{d t}=5 $$

In Exercises 1–10, use the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite integral for the given value of Round your answer to four decimal places and compare the results with the exact value of the definite integral. $$ \int_{2}^{3} \frac{2}{x^{2}} d x, \quad n=4 $$

Evaluating a Definite Integral as a Limit In Exercises \(3-8\) , evaluate the definite integral by the limit definition. $$ \int_{-1}^{1} x^{3} d x $$

In Exercises 3–6, find the general solution of the differential equation and check the result by differentiation. $$ \frac{d y}{d x}=x^{3 / 2} $$

Finding an Indefinite Integral In Exercises \(5-26\) , find the indefinite integral and check the result by differentiation. $$ \int(1+6 x)^{4}(6) d x $$

In Exercises 1–10, use the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite integral for the given value of Round your answer to four decimal places and compare the results with the exact value of the definite integral. $$ \int_{1}^{3} x^{3} d x, \quad n=6 $$

Evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{0}^{2} 6 x d x\)

Evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{-3}^{1} 8 d t\)