Home/Calculus II/Improper integrals/Infinite intervalsSection 9.1Infinite intervalsFormulavizCopy LaTeX∫a∞f(x) dx=limb→∞∫abf(x) dx\int_a^{\infty}f(x)\,dx =\lim_{b\to\infty}\int_a^b f(x)\,dx∫a∞f(x)dx=b→∞lim∫abf(x)dxView detail →FormulaCopy LaTeX∫−∞bf(x) dx=lima→−∞∫abf(x) dx\int_{-\infty}^{b}f(x)\,dx =\lim_{a\to-\infty}\int_a^b f(x)\,dx∫−∞bf(x)dx=a→−∞lim∫abf(x)dxView detail →For both infinite endpoints, we choose a point ccc:Copy LaTeX∫−∞∞f(x) dx=∫−∞cf(x) dx+∫c∞f(x) dx.\int_{-\infty}^{\infty}f(x)\,dx =\int_{-\infty}^{c}f(x)\,dx+ \int_c^{\infty}f(x)\,dx.∫−∞∞f(x)dx=∫−∞cf(x)dx+∫c∞f(x)dx.View detail →The integral converges only if both parts converge.
FormulavizCopy LaTeX∫a∞f(x) dx=limb→∞∫abf(x) dx\int_a^{\infty}f(x)\,dx =\lim_{b\to\infty}\int_a^b f(x)\,dx∫a∞f(x)dx=b→∞lim∫abf(x)dxView detail →
FormulaCopy LaTeX∫−∞bf(x) dx=lima→−∞∫abf(x) dx\int_{-\infty}^{b}f(x)\,dx =\lim_{a\to-\infty}\int_a^b f(x)\,dx∫−∞bf(x)dx=a→−∞lim∫abf(x)dxView detail →
For both infinite endpoints, we choose a point ccc:Copy LaTeX∫−∞∞f(x) dx=∫−∞cf(x) dx+∫c∞f(x) dx.\int_{-\infty}^{\infty}f(x)\,dx =\int_{-\infty}^{c}f(x)\,dx+ \int_c^{\infty}f(x)\,dx.∫−∞∞f(x)dx=∫−∞cf(x)dx+∫c∞f(x)dx.View detail →