Evaluate the definite integral. \int_{0}^{1}\frac{7}{1+x^{2}}dx

Step 1: If f(x) is a continuous function from a to b, and if F(x) is its integral, then:
${\displaystyle {\int }_a^{b}f\left(x\right)dx=F\left(x\right){\mid }_a^b=F\left(b\right)-F\left(a\right)}$
Step 2: In this case, ${\displaystyle f\left(x\right)=\frac{7}{1+x^2}}$. Find its integral.
${\displaystyle 7{\tan }^{-1}\left(x\right){\mid }_0^1}$
Step 3: SInce ${\displaystyle F\left(x\right){\mid }_a^b=F\left(b\right)-F\left(a\right)}$, expand the above into F(1)−F(0):
${\displaystyle 7{\tan }^{-1}\left(1\right)-7{\tan }^{-1}\left(0\right)}$
Step 4: Simplify.
${\displaystyle \frac{7\pi }{4}}$