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pedzenekO

Answered question

2021-06-03

Find formulas for the functions represented by the integrals.

\int_{- \frac{π}{4}}^{x} \sec^{2} 0 d 0

Aubree Mcintyre

Skilled2021-06-04Added 73 answers

Step 1
It is given that,

\int_{- \frac{π}{4}}^{x} \sec^{2} 0 d 0

We have to find formulas for the functions represented by the integrals.
Step 2
We have,

\int_{- \frac{π}{4}}^{x} \sec^{2} 0 d 0

...(1)
We know that, indefinite integral:

\int \sec^{2} x d x = \tan x + C

, where C is arbitrary constant
Then,for definite integral:

\int_{- \frac{π}{4}}^{x} \sec^{2} 0 d 0 = [\tan 0]_{- \frac{π}{4}}^{x}

\Rightarrow \int_{- \frac{π}{4}}^{x} \sec^{2} 0 d 0 = \tan x - \tan (\frac{- π}{4})

\Rightarrow \int_{- \frac{π}{4}}^{x} \sec^{2} 0 d 0 = \tan x - (- \tan (\frac{π}{4})), (s i n c e, \tan (- 0) = - \tan (0))

\Rightarrow \int_{- \frac{π}{4}}^{x} \sec^{2} 0 d 0 = \tan x + \tan (\frac{π}{4})

\Rightarrow \int_{- \frac{π}{4}}^{x} \sec^{2} 0 d 0 = \tan x + 1, (s i n c e, \tan (\frac{π}{4}) = 1)

Hence,the required formula is

\int_{- \frac{π}{4}}^{x} \sec^{2} 0 d 0 = \tan x + 1

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