91Ó°ÊÓ

use the Exponential Rule to find the indefinite integral. $$ \int e^{4 x} d x $$

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. \(\int_{0}^{2} 3 d x\)

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=\sqrt{x}, \quad[0,1] $$

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int\left(4 x^{3}-\frac{1}{x^{2}}\right) d x=x^{4}+\frac{1}{x}+C $$

identify \(u\) and \(d u / d x\) for the integral \(\int u^{n}(d u / d x) d x .\) $$ \int \sqrt{1-x^{2}}(-2 x) d x $$

use the Exponential Rule to find the indefinite integral. $$ \int e^{-0.25 x} d x $$

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. \(\int_{0}^{3} 4 d x\)

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int\left(1-\frac{1}{\sqrt[3]{x^{2}}}\right) d x=x-3 \sqrt[3]{x}+C $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=1-x^{2}, \quad[-1,1] $$

identify \(u\) and \(d u / d x\) for the integral \(\int u^{n}(d u / d x) d x .\) $$ \int\left(4+\frac{1}{x^{2}}\right)^{5}\left(\frac{-2}{x^{3}}\right) d x $$