Tugas Kalkulus

1. $\int{{{x}^{n}}{{e}^{x}}dx}$ =
u = ${{e}^{n}}\to {{u}^{1}}$ = $n{{x}^{n-1}}$
${{v}^{1}}={{e}^{x}}\to v={{e}^{x}}$
$\int {{x}^{n}}{{e}^{x}}dx={{x}^{n}}{{e}^{x}}-\int en{{x}^{n-1}}e$
$\int {{x}^{n}}{{e}^{x}}dx={{x}^{n}}{{e}^{x}}-n\int {{x}^{n-1}}{{e}^{x}}dx$


2. $\int {{\sin }^{x}}xdx=\int {{\sin }^{n-1}}xsinxdx$
$u={{\sin }^{n}}-1\to v=-{{\cos }^{x}}$
$du=(n-1)si{{n}^{n-2}}xcosxdx\to dv={{\sin }^{x}}dx$
$=\int {{\sin }^{n}}xdx=\int udv=uv-\int vdu$
$=-{{\sin }^{n-1}}xcosx-\int (n-1)(-cosx)si{{n}^{n-2}}xcosxdx$
$=-{{\sin }^{n-1}}xcosx+(n-1)\int {{\sin }^{n-2}}x(1-{{\sin }^{2}}x)dx$
$=n\int {{\sin }^{n}}xdx=-{{\sin }^{n-1}}xcosx+(n-1)\int {{\sin }^{n-2}}xdx$
$=\int {{\sin }^{n}}xdx=-\frac{1}{n}{{\sin }^{n-1}}xcosx+\frac{n-1}{n}\int si{{n}^{n-2}}xdx$
$=\int sinxdx=-cosx+c$ $\mathop{\int }^{}\cos xdx=\sin x+c$


3. $\int co{{s}^{n}}xdx=\int cos{{e}^{n-1}}xcosxdx$
$u=co{{s}^{n-1}}x\to v=sinx$
$du=(n-1)co{{s}^{n-2}}x-sinxdx\to dv=cosxdx$
$\int udv=uv-\int vdu$
$=sinxco{{s}^{n-1}}x-\int sinx(n-1)cosn-2xsinxdx$
$=sinxco{{s}^{n-1}}(x)+(n-1)\int (1-co{{s}^{2}}x)co{{s}^{n-2}}xdx$
$=sinxco{{s}^{n-1}}x+(n-1)\int co{{s}^{n-2}}xdx-(n-1)\int co{{s}^{n}}xdx$
$=sinxco{{s}^{n-1}}x+(n-1)\int cos{{e}^{n-2}}xdx$
$=\frac{1}{n}sinxco{{s}^{n-1}}x+(\frac{n-1}{n})\int co{{s}^{n-2}}xdx$


4. $\int si{{n}^{n}}xdx=\int si{{n}^{n-1}}xsinxdx$
$*\int udv$
$=uv-\int vdu$
$u=si{{n}^{n-1}}x\to dv=sinxdx$
$du=(n-1)si{{n}^{n-2}}xcosxdx\to v=-cosx$
$=si{{n}^{n-1}}-(cosx)-\int (-cosx)(n-1)si{{n}^{n-2}}xcosxdx$
$=-si{{n}^{n-1}}xcosx+(n-1)\int cos2xsi{{n}^{n-2}}xdx$
$(n-1)\int (1-sin2x)sinn-2xdx$
$(n-1)\mathop{\int }^{}({{\sin }^{n-2}}x-{{\sin }^{n}}x)dx$
$(n-1)\mathop{\int }^{}{{\sin }^{n-2}}xdx-(n-1)\mathop{\int }^{}{{\sin }^{n}}xdx$
$\mathop{\int }^{}{{\sin }^{n}}xdx+(n-1)\mathop{\int }^{}{{\sin }^{n}}xdx$
$=-{{\sin }^{n-1}}x\cos x+(n-1)\mathop{\int }^{}{{\sin }^{n-2}}xdx-(n-1)\mathop{\int }^{}{{\sin }^{n}}xdx$
$=-{{\sin }^{n-1}}x\cos x+(n-1)\mathop{\int }^{}{{\sin }^{n-2}}xdx+(n-1)\mathop{\int }^{}{{\sin }^{n}}xdx$
$n\mathop{\int }^{}{{\sin }^{n}}xdx=-{{\sin }^{n-1}}x\cos x+(n-1)\mathop{\int }^{}{{\sin }^{n-2}}xdx$
${{\sin }^{n}}xdx=\frac{-{{\sin }^{n-1}}x\cos x}{n}+\frac{n-1}{n}\mathop{\int }^{}{{\sin }^{n-2}}xdx$


5. $\mathop{\int }^{}{{\cos }^{n}}xdx={{\cos }^{n-1}}x\sin x-\mathop{\int }^{}\sin xd({{\cos }^{n-1}}x)$
$={{\cos }^{n-1}}x\sin x+(n-1)\mathop{\int }^{}\sin x{{\cos }^{n-2}}{{\sin }^{2}}xdx$
$={{\cos }^{n-1}}x\sin x+(n-1)\mathop{\int }^{}{{\cos }^{n-2}}x{{\sin }^{2}}xdx$
$={{\cos }^{n-1}}x\sin x+(n-1)\mathop{\int }^{}{{\cos }^{n-2}}x(1-{{\cos }^{2}}x)dx$
$={{\cos }^{n-1}}x\sin x+(n-1)\mathop{\int }^{}{{\cos }^{n-2}}xdx-(n-1)\mathop{\int }^{}{{\cos }^{n}}xdx$
$={{\cos }^{n-1}}x\sin x+(n-1)I{{n}_{2}}-(n-1)$