Integral substitusi bentuk akar

Kalkulus Matematika · SMA medium

Hasil $\displaystyle\int \dfrac{3x^2}{\sqrt{2x^3+4}}\,dx = \ldots$

Kunci jawaban

Misal $u=2x^3+4$ , $du=6x^2dx$ . $\int u^{-1/2}\tfrac{du}{2}=\sqrt u=\sqrt{2x^3+4}+C$ .

(Jawaban dihitung & diverifikasi dengan Python/SymPy.)

Substitusi $u=2x^3+4 \Rightarrow du=6x^2\,dx \Rightarrow 3x^2\,dx=\tfrac12du$ .

$\int \dfrac{3x^2}{\sqrt{2x^3+4}}\,dx=\int \dfrac{1}{\sqrt u}\cdot\tfrac12\,du=\tfrac12\cdot2\sqrt u=\sqrt{2x^3+4}+C$

→ C.