Homework $5-$ Additional Exercises

Problems $1-3$ regard trigonometric integrals $($see section $7\mathrm{.}2)$ and problems $4-8$ regard trigonometric substitutions $($see section $7\mathrm{.}3).$

Evaluate ${\displaystyle \underset{\frac{}{2}}{\overset{3\frac{}{4}}{}}}si{n}^{5}x*co{s}^{3}xdx.$

Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}ta{n}^{2}x*secxdx.$

Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{cosx+sinx}{sin2x}dx.$

Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{1}{x^2\sqrt[\phantom{2}]{x^2-9}}dx$ using the substitution: $x=3sec.$

Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{1}{\sqrt[\phantom{2}]{x^2+16}}dx.$

Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}x\sqrt[\phantom{2}]{1-x^4}dx.$

Prove the formula $A=\frac{1}{2}{r}^2$ for the area of a sector of a circle with radius $r$ and central angle $.$

Hint:

Assume that and place the center of the circle at the origin so that it has the equation: $x^2+y^2=r^2.$

Now

It is easy to find the area of the triangle.

Show that

Area$(SegmentPQR)={\displaystyle \underset{rcos}{\overset{r}{}}}\sqrt[\phantom{2}]{r^2-x^2}dx$

and make the substitution $x=rcos$ to transform the problem into a definite integral of a trig function of $.$ I NEED $1,3,5,6,7$ ONLY