Fun Learning

Di bawah ini adalah tugas kalkulus ke-4 yang sudah saya kerjakan di Mata Kuliah Kalkulus 2, tentang Integral Parsial. Oleh dosen saya, yaitu Bpk. Suwarto, M.Pd dengan kode kelas MT112A.

Tugas Kalkulus

$1. \int cos^{n} xdx$

Misal :

$\int cos^{n}xdx = \int cos^{n-1} . cos x$ .

$\int u.dv = uv-\int v.du$ .

$u = cos^{n-1} x$ .

$du = \left ( n-1 \right )cos^{n-2}x . -sinx$ .

$dv = cos x dx$ .

$v = sin x$ .

$\int cos^{n-1} x cos x dx = cos^{n-1}x.sin x-\int sin x.\left ( n-1 \right ) cos^{n-2}x.-sin x dx$ .

$= cos^{n-1}x . sin x + \int \left ( n-1 \right )cos^{n-1} x . sin^{2}xdx$ .

$sin^{2}x + cos^{2}x = 1\rightarrow sin^{2}x = \left ( 1-cos^{2}x \right )dx$ .

$sin^{2}x + cos^{2}x + \int \left ( n-1 \right )cos^{n-2}x \left ( 1-cos^{2}x \right ) dx$ .

$\int \left ( n-1 \right )cos^{2}x - \left ( n-1 \right )cos^{2}x . cos^{2}x$ .

$\int \left ( n-1 \right )cos^{n-2}x - \left ( n-1 \right )cos^{n}x . cos^{2}xdx$ .

$\int cos^{n}xdx = cos^{n-1}x.sin x + \int \left ( n-1 \right )cos^{n-2}xdx-\int \left ( n-1 \right )cos^{n}xdx$ .

$1\int cos^{n}xdx + \int \left ( n-1 \right )\int cos^{n}xdx$ .

$\frac{n}{n}\int cos^{n}xdx = cos^{n-1} x.sin x+\left ( n-1 \right )cos^{n-2}xdx$ .

$\int cos^{n}xdx = \frac{1}{n}cos^{n-1} x.sin x + \frac{n-1}{n}\int cos^{n-2}xdx$ .

$2. \int sin^{n}xdx$ .

$\int sin^{n}xdx = \frac{-1}{n}cosx . sin^{n-1}x + \frac{n-1}{n}$ .

$\int sin^{n-2}xdx$ .

Misal :

$u=sin^{n-1}x\rightarrow du=\left ( n-1 \right ).sin^{n-2}x . cos xdx$ .

$v= -cos x\rightarrow dv=sinxdx$ .

$\int sin^{n}xdx=sin\left ( n-1 \right ). sinxdx$ .

$\int sin^{n}xdx=sin^{n-1}. -cosx-\int -cos x . \left ( n-1 \right ). sin^{n-2}x . cos xdx$ .

$\int sin^{n}xdx=-cos x . sin^{n-1}x + \left ( n-1 \right )\int sin^{n-2}x.cos xdx$ .

$\int sin^{n}xdx=-cos x . sin^{n-1}x + \left ( n-1 \right )\int sin^{n-2}x . \left ( 1-sin^{2}x \right )dx$ .

$\int sin^{n}xdx=-cos x . sin^{n-1}x + \left ( n-1 \right )\int sin^{n-2}x . \left (sin^{2}x \right )dx - \left ( n-1 \right )\int sin^{n}dx$ .

$\int sin^{n}xdx= \frac{-1}{n}cos x . sin^{n-1}x + \frac{n-1}{n}\int sin^{n-2}xdx$ .

$\int cos^{-n}xdx = \frac{1}{n}cos^{-n+1}x.sinx+\int cosxdx+\frac{\left ( -n+1 \right )}{n}\int cosxdx$

$3. \int cos^{-n}xdx$ .

Misal :

$u=cos^{-n+1}x\rightarrow du=\left ( n-1 \right )cos^{-n}x.-sin xdx$ .

$dv=cosxdx\rightarrow v=sinx$ .

$\int u.dv=uv-\int v.du$ .

$\int cos^{-n+1}xcosxdx = cos^{-n+1}x.sinx-\int .\left ( -n+1 \right )cos^{-n}x.sinxdx$ .

$\int cos^{-n+1}xcosxdx = cos^{-n+1}x.sinx+\int\left ( -n+1 \right )cos^{-n}x.sin^{2}xdx$ .

$= cos^{-n+1}x.sinx+\int\left ( -n+1 \right )cos^{-n}x\left ( 1-c0s^{2}x \right )dx$ .

$= cos^{-n+1}x.sinx+\int\left ( -n+1 \right )cos^{-n}x-\left ( -n+1 \right )cos^{-n}xdx$ .

$= cos^{-n+1}x.sinx+\int\left ( -n+1 \right )cos^{-n}x-\left ( -n+1 \right )cosxdx$ .

$\int cos^{-n}xdx = cos^{-n+1}x . sin x + \int \left ( -n+1 \right )cos^{-n}xdx-\int \left ( -n+1 \right )cosxdx$ .

$1\int cos^{-n}xdx + \left ( -n+1 \right )cos^{-n}xdx$ .

$\int cos^{-n}xdx \approx cos^{-n}xdx$ .

$\frac{n}{n}\int cos^{-n}xdx = cos^{-n+1}x.sinx+\left ( -n+1 \right )\int cosxdx$

$4.\int sin^{-n}xdx$ .

Misal :

$\int sin^{-n}xdx= \int sin^{-n+1}x . sinx$ .

$u=sin^{-n+1}x \rightarrow du=\left ( -n+1 \right )sin^{-n}.cosxdx$ .

$dv=sinxdx \rightarrow v=-cosx$ .

$\int u.dv= uv-\int v.du$ .

$\int sin^{-n+1}xsinxdx= sin^{-n+1}x.-cosx-\int -cosx.\left ( -n+1 \right )sin^{-n}x.cosxdx$ .

$\int sin^{-n+1}xsinxdx= sin^{-n+1}x.-cosx+\left ( -n+1 \right )\int cos^{2}x . sin^{-n}xdx$ .

$\int sin^{-n+1}x . sinxdx = sin^{-n+1}x . -cosx + \left ( -n+1 \right )\int \left ( 1-sin^{2}x \right )sin^{-n}xdx$ .

$\int sin^{-n+1}x . sinxdx = sin^{-n+1}x . -cosx + \left ( -n+1 \right )\int sin^{-n}x-sin^{-2}xdx$ .

$\int sin^{-n+1}x . sinxdx = sin^{-n+1}x . -cosx + \left ( -n+1 \right )\int sin^{-n}xdx-\left ( -n+1 \right )\int sin^{-n-2}xdx$ .

$1\int sin^{-n}xdx +\left ( -n+1 \right )\int sin^{-n} xdx$ .

$\int sin^{-n}xdx \approx \int sin^{-n} xdx$ .

$\frac{n}n{}\int sin^{-n}xdx = -sin^{-n+1}x . cosx + \left ( -n+1 \right )\int sin^{-n-2}xdx$ .

$\int sin^{-n}xdx = -\frac{1}{n}sin^{-n+1}x . cosx +\frac{\left ( -n+1 \right )}{n}\int sin^{-n-2}xdx$

Demikian tugas yang telah saya selesaikan, mohon maaf bila ada kekurangan atau kesalahan dalam penghitungannya. Silahkan beri saran, kritik untuk pos saya agar lebih baik lagi. Sekian dan Terima kasih 🙂

Salam Hangat,

Evi S.

S	S	R	K	J	S	M
1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30

Fun Learning

Tugas Kalkulus Ke-4

Proyek

Pengalaman Diri

Tentang Penulis

Fun Learning