Chapter 11

Find the indefinite integral. $$\int 4 e^{x} d x$$

Sketch the graph and find the area of the region bounded by the graph of the function \(f\) and the lines \(y=0, x=a\), and \(x=b\) $$f(x)=x^{2}-2 x ; a=-1, b=1$$

Evaluate the definite integral. $$\int_{1}^{2} \frac{\ln x}{x} d x$$

Evaluate the definite integral. $$\int_{1}^{2}\left(t^{5}-t^{3}+1\right) d t$$

Find the indefinite integral. $$\int \frac{e^{2 x}}{1+e^{2 x}} d x$$

Find the indefinite integral. $\int\left(1+e^{x}\right) d x$$

Sketch the graph and find the area of the region bounded by the graph of the function \(f\) and the lines \(y=0, x=a\), and \(x=b\) $$f(x)=-x^{2}+4 x-3 ; a=-1, b=2$$

Find the area of the region under the graph of \(f\) on \([a, b]\). $$f(x)=x^{2}-2 x+2 ;[-1,2]$$

Evaluate the definite integral. $$\int_{2}^{4} \frac{1}{x} d x$$

Find the indefinite integral. $$\int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x$$

Find the indefinite integral. $$\int\left(1+x+e^{x}\right) d x$$

Sketch the graph and find the area of the region bounded by the graph of the function \(f\) and the lines \(y=0, x=a\), and \(x=b\) $$f(x)=x^{3}-x^{2} ; a=-1, b=1$$

Evaluate the definite integral. $$\int_{1}^{3} \frac{2}{x} d x$$

Find the indefinite integral. $$\int \frac{e^{-1 / x}}{x^{2}} d x$$

Find the indefinite integral. $$\int\left(2+x+2 x^{2}+e^{x}\right) d x$$

Sketch the graph and find the area of the region bounded by the graph of the function \(f\) and the lines \(y=0, x=a\), and \(x=b\) $$f(x)=x^{3}-4 x^{2}+3 x ; a=0, b=2$$

Find the area of the region under the graph of \(f\) on \([a, b]\). $$f(x)=\frac{1}{x^{2}} ;[1,2]$$

Evaluate the definite integral. $$\int_{0}^{4} x\left(x^{2}-1\right) d x$$

Find the indefinite integral. $$\int \frac{e^{3 x}+x^{2}}{\left(e^{3 x}+x^{3}\right)^{3}} d x$$

Find the indefinite integral. $$\int\left(4 x^{3}-\frac{2}{x^{2}}-1\right) d x$$

Sketch the graph and find the area of the region bounded by the graph of the function \(f\) and the lines \(y=0, x=a\), and \(x=b\) $$f(x)=4 x^{1 / 3}+x^{4 / 3} ; a=-1, b=8$$

Find the area of the region under the graph of \(f\) on \([a, b]\). $$f(x)=2+\sqrt{x+1} ;[0,3]$$

Evaluate the definite integral. $$\int_{0}^{2}(x-4)(x-1) d x$$

Find the indefinite integral. $$\int \frac{e^{x}-e^{-x}}{\left(e^{x}+e^{-x}\right)^{3 / 2}} d x$$

Find the indefinite integral. $$\int\left(6 x^{3}+\frac{3}{x^{2}}-x\right) d x$$

Sketch the graph and find the area of the region bounded by the graph of the function \(f\) and the lines \(y=0, x=a\), and \(x=b\) $$f(x)=e^{x}-1 ; a=-1, b=3$$

Find the area of the region under the graph of \(f\) on \([a, b]\). $$f(x)=e^{-x / 2} ;[-1,2]$$

Evaluate the definite integral. $$\int_{1}^{3}\left(t^{2}-t\right)^{2} d t$$

Find the indefinite integral. $$\int e^{2 x}\left(e^{2 x}+1\right)^{3} d x$$

Find the indefinite integral. $$\int\left(x^{5 / 2}+2 x^{3 / 2}-x\right) d x$$

Sketch the graph and find the area of the region bounded by the graph of the function \(f\) and the lines \(y=0, x=a\), and \(x=b\) $$f(x)=x e^{x^{2}} ; a=0, b=2$$

Find the area of the region under the graph of \(f\) on \([a, b]\). $f(x)=\frac{\ln x}{4 x} ;[1,2]$$

Evaluate the definite integral. $$\int_{-1}^{1}\left(x^{2}-1\right)^{2} d x$$

Find the indefinite integral. $$\int e^{-x}\left(1+e^{-x}\right) d x$$

Find the indefinite integral. $$\int\left(t^{3 / 2}+2 t^{1 / 2}-4 t^{-1 / 2}\right) d t$$

Sketch the graph and find the area of the region completely enclosed by the graphs of the given functions \(f\) and \(g\). $$f(x)=x+2\( and \)g(x)=x^{2}-4$$

Evaluate the definite integral. $$\int_{-3}^{-1} \frac{1}{x^{2}} d x$$

Find the indefinite integral. $$\int \frac{\ln 5 x}{x} d x$$

Find the indefinite integral. $$\int\left(\sqrt{x}+\frac{3}{\sqrt{x}}\right) d x$$

Sketch the graph and find the area of the region completely enclosed by the graphs of the given functions \(f\) and \(g\). $$f(x)=-x^{2}+4 x\( and \)g(x)=2 x-3$$

Find the average value of the function f over the indicated interval \([a, b]\). $$f(x)=8-x ;[1,4]$$

Evaluate the definite integral. $$\int_{1}^{2} \frac{2}{x^{3}} d x$$

Find the indefinite integral. $$\int \frac{(\ln u)^{3}}{u} d u$$

Find the indefinite integral. $$\int\left(\sqrt[3]{x^{2}}-\frac{1}{x^{2}}\right) d x$$

Sketch the graph and find the area of the region completely enclosed by the graphs of the given functions \(f\) and \(g\). $$f(x)=x^{2}\( and \)g(x)=x^{3}$$

Find the average value of the function f over the indicated interval \([a, b]\). $$f(x)=2 x^{2}-3 ;[1,3]$$

Evaluate the definite integral. $$\int_{1}^{4}\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right) d x$$

Find the indefinite integral. $$\int \frac{1}{x \ln x} d x$$

Find the indefinite integral. $$\int\left(\frac{u^{3}+2 u^{2}-u}{3 u}\right) d u$$

Sketch the graph and find the area of the region completely enclosed by the graphs of the given functions \(f\) and \(g\). $$f(x)=x^{3}+2 x^{2}-3 x\( and \)g(x)=0$$