# Find formulas for the functions represented by the integrals. \int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0

Applications of integrals
Find formulas for the functions represented by the integrals.
$$\int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0$$

2021-06-04
Step 1
It is given that, $$\int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0$$
We have to find formulas for the functions represented by the integrals.
Step 2
We have, $$\int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0$$...(1)
We know that, indefinite integral: $$\int \sec^{2}xdx=\tan x + C$$, where C is arbitrary constant
Then,for definite integral:
$$\int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0=[\tan 0]_{-\frac{\pi}{4}}^{x}$$
$$\Rightarrow \int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0=\tan x-\tan(\frac{-\pi}{4})$$
$$\Rightarrow \int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0=\tan x - (-\tan(\frac{\pi}{4})), (since, \tan(-0)=-\tan (0))$$
$$\Rightarrow \int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0=\tan x + \tan(\frac{\pi}{4})$$
$$\Rightarrow \int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0=\tan x + 1, (since, \tan(\frac{\pi}{4})=1)$$
Hence,the required formula is
$$\int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0=\tan x + 1$$