Lutasin ang ∫ (from 0 to pi) of cos^3θsinphidphi

$$\int _ { 0 } ^ { \pi } \cos ^ { 3 } \theta \sin \phi d \phi$$

I-evaluate

$$2\left(\cos(\theta )\right)^{3}$$

Mga Hakbang ng Solution

Suriin muna ang indefinite na integral.

$$\int \left(\cos(\theta )\right)^{3}\sin(ϕ)\mathrm{d}ϕ$$

I-factour out ang constant gamit ang \int af\left(ϕ\right)\mathrm{d}ϕ=a\int f\left(ϕ\right)\mathrm{d}ϕ.

$$\left(\cos(\theta )\right)^{3}\int \sin(ϕ)\mathrm{d}ϕ$$

Gamitin ang \int \sin(\theta )\mathrm{d}\theta =-\cos(\theta ) mula sa talahanayan ng mga karaniwang integral upang makuha ang resulta.

$$-\left(\cos(\theta )\right)^{3}\cos(ϕ)$$

Ang definite integral ay ang antiderivative ng expression na na-evaluate sa upper limit ng integration minus ang antiderivative na na-evaluate sa lower limit ng integration.

$$-\left(\cos(\theta )\right)^{3}\cos(\pi )+\left(\cos(\theta )\right)^{3}\cos(0)$$

Pasimplehin.

$$2\left(\cos(\theta )\right)^{3}$$

I-differentiate ang w.r.t. θ

$$-6\sin(\theta )\left(\cos(\theta )\right)^{2}$$