Question

Find the integrals of the functions in Exercises 1 to 22
1. ${\sin }^3\left(2x+1\right)$
2. ${\sin }^{3}x{\cos }^{3}x$
3.$\frac{\cos x-\sin x}{1+\sin 2x}$

Answer 1

$\left(1\right)si{n}^3(2x+1)dx$

$\int sin(2x+1)(1-co{s}^2(2x+1\left)\right)dx$

Let $cos(2x+1)=t$

$-2sin(2x+1)dx=dt$

$=\frac{-1}{2}\int (1-t^2)dt$

$=-\frac{1}{2}[t-\frac{t^3}{3}]+c\Rightarrow$

$=-\frac{1}{2}\left[cos\right(2x+1)-\frac{co{s}^3(2x+1)}{3}]+c$

(2) $\int si{n}^{3}xco{s}^{3}xdx=?$

$=\int sinx(1-co{s}^{2}x)co{s}^{3}xdx$

Let cos x= t

- sin x dx = dt

$=-\int (1-t^2)t^{3}dt=-\frac{t^4}{4}+\frac{t^6}{6}+c$

$=-\frac{co{s}^{4}x}{4}+\frac{co{s}^{6}x}{6}+c$

(3) $\int \frac{cosx-sinx}{1+sin2x}dx=\frac{cosx-sinx}{(cosx+sinx{)}^2}dx$

Let sin x+ cos x = t

(cos x - sin x) dx= dt

$=\int \frac{dt}{t^2}=-\frac{1}{t}+c=-\frac{1}{sinx+cosx}+c$