Exercises on the Properties of Definite Integrals

Prerequisite knowledge: some properties of definite integrals

Exercise 1

计算${\int }_{-2}^2(\frac{x\cos x}{1+x^2}+\sqrt{4-x^2})dx$

Solution:
\qquadBecause $f(x)=\frac{x\cos x}{1+x^2}$is an odd function

\qquad So ${\int }_{-2}^2\frac{x\cos x}{1+x^2}dx=0$

\qquad原式$={\int }_{-2}^2(0+\sqrt{4-x^2})dx={\int }_{-2}^2\sqrt{4-x^2}dx=2p.m$

Exercise 2

Known $f$ in$[0,4]$ is$f(x)=|x-2|$ , and$f$ is$4$ is a periodic function, find${\int }_3^{7}f(x)dx$

Solution:
\qquadbecause $f$ is$4$ is a periodic function

\qquad So the original formula $={\int }_0^{4}f\left(x\right)dx={\int }_0^2(2-x)dx+{\int }_2^4(x-2)dx=2+2=4$