Question

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

Applications of integrals
ANSWERED
asked 2021-06-03
Find formulas for the functions represented by the integrals.
\(\int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0\)

Answers (1)

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