University⏱ 4 min

U-Substitution

Integration by substitution

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f(x)F(x)

u-substitúcia:

u=2x,  du=2dxu=2x,\;du=2dx
−12cos⁡u+C-\frac{1}{2}\cos u+C
∫sin⁡(2x) dx=−cos⁡2x2+C\int\sin(2x)\,dx = -\frac{\cos 2x}{2}+C

Substitution u = inner function simplifies the integral. Green curve = antiderivative. Differentiating green gives blue.