studymathwithstudent: INTEGRAL PARSIAL

Rumus integral parsial:

$\int udv=u.v-\int vdu$

Keterangan:

u = f(x), maka du = f(x) dx

dv = g(x) dx, maka v = g(x) dx

Contoh soal:

1. Nilai dari $\int x\sqrt{x+2}dx = ...$

Pembahasan:

Misalkan:

$u = x\to du = dx$

$dv = \sqrt{x+2}dx\to v = \int \sqrt{x+2}dx = \frac{2}{3}(x+2)^{\frac{3}{2}}dx$

Sehingga

$\int udv = u.v - \int vdu$

$\int x\sqrt{x+2} = x.\frac{2}{3}(x+2)^{\frac{3}{2}} - \int \frac{2}{3}(x+2)^{\frac{3}{2}}dx$

$\int x\sqrt{x+2} = x.\frac{2}{3}(x+2)^{\frac{3}{2}} - \frac{2}{3}.\frac{2}{5}(x+2)^{\frac{5}{2}} + C$

$\int x\sqrt{x+2} = \frac{2}{3}(x+2)^{\frac{3}{2}}\left\{ x - \frac{2}{5}(x+2)\right\} + C$

$\int x\sqrt{x+2} = \frac{2}{3}(x+2)^{\frac{3}{2}}\left\{\frac{5x - 2 (x+2)}{5} \right\} + C$

$\int x\sqrt{x+2} = \frac{2}{3}(x+2)^{\frac{3}{2}}(\frac{3x - 2}{5}) + C$ $\int x\sqrt{x+2} = \frac{2}{15}(3x - 2)(x + 2)\sqrt{x + 2} + C$

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