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

\(\int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0=\tan x + 1\)

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

\(\int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0=\tan x + 1\)