3.5 Other strategies for integration

${\displaystyle \underset{0}{\overset{4}{\int }}\frac{x}{\sqrt{1+2x}}dx}$

${\displaystyle \int \frac{x+3}{x^2+2x+2}dx}$

${\displaystyle \int x^3\sqrt{1+2x^2}dx}$

${\displaystyle \int \frac{1}{\sqrt{x^2+6x}}dx}$

${\displaystyle \int \frac{x}{x+1}dx}$

${\displaystyle \int x·2^{x^2}dx}$

${\displaystyle \int \frac{1}{4x^2+25}dx}$

${\displaystyle \int \frac{dy}{\sqrt{4-y^2}}}$

${\displaystyle \int {\text{sin}}^3(2x)\text{cos}(2x)dx}$

${\displaystyle \int \text{csc}(2w)\text{cot}(2w)dw}$

${\displaystyle \int 2^{y}dy}$

${\displaystyle {\int }_0^1\frac{3xdx}{\sqrt{x^2+8}}}$

${\displaystyle {\int }_{\mathrm{-1}\text{/}4}^{1\text{/}4}{\text{sec}}^2(\pi x)\text{tan}\left(\pi x\right)dx}$

${\displaystyle {\int }_0^{\pi \text{/}2}{\text{tan}}^2\left(\frac{x}{2}\right)dx}$

${\displaystyle \int {\text{cos}}^{3}xdx}$

${\displaystyle \int {\text{tan}}^5\left(3x\right)dx}$

${\displaystyle \int {\text{sin}}^{2}y{\text{cos}}^{3}ydy}$

[T] ${\displaystyle \int \frac{dw}{1+\text{sec}\left(\frac{w}{2}\right)}}$

[T] ${\displaystyle \int \frac{dw}{1-\text{cos}\left(7w\right)}}$

[T] ${\displaystyle {\int }_0^t\frac{dt}{4\text{cos}t+3\text{sin}t}}$

[T] ${\displaystyle \int \frac{\sqrt{x^2-9}}{3x}dx}$

[T] ${\displaystyle \int \frac{dx}{x^{1\text{/}2}+x^{1\text{/}3}}}$

[T] ${\displaystyle \int \frac{dx}{x\sqrt{x-1}}}$

[T] ${\displaystyle \int x^3\text{sin}xdx}$

[T] ${\displaystyle \int x\sqrt{x^4-9}dx}$

[T] ${\displaystyle \int \frac{x}{1+e^{\text{−}{x}^2}}dx}$

[T] ${\displaystyle \int \frac{\sqrt{3-5x}}{2x}dx}$

[T] ${\displaystyle \int \frac{dx}{x\sqrt{x-1}}}$

[T] ${\displaystyle \int e^x{\text{cos}}^{\mathrm{-1}}(e^x)dx}$

[T] ${\displaystyle {\int }_0^{\pi \text{/}4}\text{cos}(2x)dx}$

[T] ${\displaystyle {\int }_0^{1}x·e^{\text{−}{x}^2}dx}$

[T] ${\displaystyle {\int }_0^8\frac{2x}{\sqrt{x^2+36}}dx}$

[T] ${\displaystyle {\int }_0^{2\text{/}\sqrt{3}}\frac{1}{4+9x^2}dx}$

[T] ${\displaystyle \int \frac{dx}{x^2+4x+13}}$

[T] ${\displaystyle \int \frac{dx}{1+\text{sin}x}}$

${\displaystyle \int \frac{dx}{x^2+2x+10}}$

${\displaystyle \int \frac{dx}{\sqrt{x^2-6x}}}$

${\displaystyle \int \frac{e^x}{\sqrt{e^{2x}-4}}dx}$

${\displaystyle \int \frac{\text{cos}x}{{\text{sin}}^{2}x+2\text{sin}x}dx}$

${\displaystyle \int \frac{\text{arctan}\left(x^3\right)}{x^4}dx}$

${\displaystyle \int \frac{\text{ln}\left|x\right|\text{arcsin}\left(\text{ln}\left|x\right|\right)}{x}dx}$

${\displaystyle \int \frac{dx}{\sqrt{x^2+16}}}$

${\displaystyle \int \frac{3x}{2x+7}dx}$

${\displaystyle \int \frac{dx}{1-\text{cos}\left(4x\right)}}$

${\displaystyle \int \frac{dx}{\sqrt{4x+1}}}$

Find the area bounded by $y\left(4+25x^2\right)=5,x=0,y=0,\text{and}x=4.$ Use a table of integrals or a CAS.

The region bounded between the curve $y=\frac{1}{\sqrt{1+\text{cos}x}},0.3\le x\le 1.1,$ and the x -axis is revolved about the x -axis to generate a solid. Use a table of integrals to find the volume of the solid generated. (Round the answer to two decimal places.)

Use substitution and a table of integrals to find the area of the surface generated by revolving the curve $y=e^x,0\le x\le 3,$ about the x -axis. (Round the answer to two decimal places.)

[T] Use an integral table and a calculator to find the area of the surface generated by revolving the curve $y=\frac{x^2}{2},0\le x\le 1,$ about the x -axis. (Round the answer to two decimal places.)

[T] Use a CAS or tables to find the area of the surface generated by revolving the curve $y=\text{cos}x,0\le x\le \frac{\pi }{2},$ about the x -axis. (Round the answer to two decimal places.)

Find the length of the curve $y=\frac{x^2}{4}$ over $\left[0,8\right].$

Find the length of the curve $y=e^x$ over $\left[0,\text{ln}(2)\right].$

Find the area of the surface formed by revolving the graph of $y=2\sqrt{x}$ over the interval $\left[0,9\right]$ about the x -axis.

Find the average value of the function $f(x)=\frac{1}{x^2+1}$ over the interval $\left[\mathrm{-3},3\right].$

Approximate the arc length of the curve $y=\text{tan}\left(\pi x\right)$ over the interval $\left[0,\frac{1}{4}\right].$ (Round the answer to three decimal places.)