Difference between revisions of "Nabla integral"
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<strong>Theorem:</strong> The following formula holds: | <strong>Theorem:</strong> The following formula holds: | ||
$$\int_a^b f(t)\nabla t = \int_a^c f(t) \nabla t +\int_c^b f(t) \nabla t$$ | $$\int_a^b f(t)\nabla t = \int_a^c f(t) \nabla t +\int_c^b f(t) \nabla t$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem (Integration by parts,I):</strong> The following formula holds: | ||
| + | $$\int_a^b f(t)g^{\nabla}(t) \nabla t = (fg)(b)-(fg)(a)-\int_a^b f^{\nabla}(t)g(\rho(t)) \nabla t.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem (Integration by parts,II):</strong> The following formula holds: | ||
| + | $$\int_a^b f(\rho(t))g^{\nabla}(t) \nabla t = (fg)(b)-(fg)(a)-\int_a^b f^{\nabla}(t)g(t) \nabla t.$$ | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 16:41, 20 October 2014
Theorem: The following formula holds: $$\int_a^b f(t)+g(t) \nabla t = \int_a^b f(t) \nabla t + \int_a^b g(t) \nabla t$$
Proof: █
Theorem: The following formula holds: $$\int_a^b \alpha f(t) \nabla t = \alpha \int_a^b f(t) \nabla t$$
Proof: █
Theorem: The following formula holds: $$\int_a^b f(t) \nabla t = -\int_b^a f(t) \nabla t$$
Proof: █
Theorem: The following formula holds: $$\int_a^b f(t)\nabla t = \int_a^c f(t) \nabla t +\int_c^b f(t) \nabla t$$
Proof: █
Theorem (Integration by parts,I): The following formula holds: $$\int_a^b f(t)g^{\nabla}(t) \nabla t = (fg)(b)-(fg)(a)-\int_a^b f^{\nabla}(t)g(\rho(t)) \nabla t.$$
Proof: █
Theorem (Integration by parts,II): The following formula holds: $$\int_a^b f(\rho(t))g^{\nabla}(t) \nabla t = (fg)(b)-(fg)(a)-\int_a^b f^{\nabla}(t)g(t) \nabla t.$$
Proof: █
Theorem: The following formula holds: $$\int_a^a f(t) \nabla t = 0$$
Proof: █
Theorem (Fundamental theorem of calculus): The following formula holds: $$\int_a^b f^{\nabla}(t) \nabla t = f(b)-f(a).$$
Proof: █
