Question

Suppose f(x) is continuous on R and that for some real number c, both

${\int }_{-\infty }^{C}f\left(x\right)dx+{\int }_c^{\infty }f\left(x\right)dx$ exist. Use properties of definite integrals to prove that for all real numbers d,${\int }_{-\infty }^{C}f\left(x\right)dx+{\int }_c^{\infty }f\left(x\right)dx$is equal to${\int }_{-\infty }^{d}f\left(x\right)dx+{\int }_d^{\infty }f\left(x\right)dx.$

Short answer

The given statement is proved.

Step-by-step solution

Step 1. Given Information.

The given integral is${\int }_{-\infty }^{C}f\left(x\right)dx+{\int }_c^{\infty }f\left(x\right)dx.$

Step 2. Prove.

To prove that for all real numbers d,${\int }_{-\infty }^{C}f\left(x\right)dx+{\int }_c^{\infty }f\left(x\right)dx$is equal to ${\int }_{-\infty }^{d}f\left(x\right)dx+{\int }_d^{\infty }f\left(x\right)dx.$ Let the function f is continuous on [a,b] and for any real number c,${\int }_a^{b}f\left(x\right)dx={\int }_a^{C}f\left(x\right)dx+{\int }_c^{b}f\left(x\right)dx.$

We will prove the given statements in two cases when c < d and whend < c.

Step 3. Estimate when c < d.

We will use the property of definite integrals,

$$\begin{gathered} {\int }_{-\infty }^{C}f\left(x\right)dx+{\int }_c^{\infty }f\left(x\right)dx \\ ={\int }_{-\infty }^{C}f\left(x\right)dx+{\int }_c^{d}f\left(x\right)dx+{\int }_d^{\infty }f\left(x\right)dx \\ \mathrm{Use}{\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}={\int }_{\mathrm{a}}^{\mathrm{C}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+{\int }_{\mathrm{c}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ ={\int }_{-\infty }^{\mathrm{C}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+{\int }_{\mathrm{c}}^{\infty }\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \end{gathered}$$

Step 4. Estimate when d < c.

We will use the property of definite integrals,

$$\begin{gathered} {\int }_{-\infty }^{d}f\left(x\right)dx+{\int }_d^{\infty }f\left(x\right)dx \\ ={\int }_{-\infty }^{d}f\left(x\right)dx+{\int }_d^{c}f\left(x\right)dx+{\int }_c^{\infty }f\left(x\right)dx \\ \mathrm{Use}{\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}={\int }_{\mathrm{a}}^c\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+{\int }_{\mathrm{c}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ ={\int }_{-\infty }^{\mathrm{C}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+{\int }_{\mathrm{c}}^{\infty }\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \end{gathered}$$

Thus, role="math" localid="1649143526896" ${\int }_{-\infty }^{c}f\left(x\right)dx+{\int }_c^{\infty }f\left(x\right)dx={\int }_{-\infty }^{d}f\left(x\right)dx+{\int }_d^{\infty }f\left(x\right)dx.$

Hence proved.