Tama-kalkulus

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=u...