91Ó°ÊÓ

Find the indefinite integral and check the result by differentiation. $$ \int \frac{x}{\sqrt{1-x^{2}}} d x $$

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{1}^{4}(3-|x-3|) d x $$

Find the indefinite integral and check the result by differentiation. $$ \int \frac{x^{3}}{\sqrt{1+x^{4}}} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int\left(1+\frac{1}{t}\right)^{3}\left(\frac{1}{t^{2}}\right) d t $$

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

Use the error formulas in Theorem \(4.19\) to estimate the error in approximating the integral, with \(n=4\), using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{0}^{1} \frac{1}{x+1} d x $$

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{0}^{3}\left|x^{2}-4\right| d x $$

Sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral \((a>0, r>0)\). $$ \int_{0}^{4} \frac{x}{2} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int\left[x^{2}+\frac{1}{(3 x)^{2}}\right] d x $$

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{0}^{4}\left|x^{2}-4 x+3\right| d x $$