What is the integral of 1/(sin(x+pi/4)) ?

The integral of 1/(sin(x+pi/4)) is ln|tan(1/8 (4x+pi))|+C

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integral of 1/(sin(x+pi/4))

\int \frac{1}{sin ( x + \frac{π}{4} )} d x

ln tan (\frac{1}{8} (4 x + π)) + C

Solution steps

\int \frac{1}{sin ( x + \frac{π}{4} )} d x

Apply u-substitution

: \int csc (u) d u

= \int csc (u) d u

Apply u-substitution

: \int \frac{1}{v} d v

= \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= ln ∣ v ∣

Substitute back

= ln tan (\frac{x + \frac{π}{4}}{2})

Simplify

ln tan (\frac{x + \frac{π}{4}}{2}) : ln tan (\frac{1}{8} (4 x + π))

= ln tan (\frac{1}{8} (4 x + π))

Add a constant to the solution

= ln tan (\frac{1}{8} (4 x + π)) + C

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What is the integral of 1/(sin(x+pi/4)) ?
The integral of 1/(sin(x+pi/4)) is ln|tan(1/8 (4x+pi))|+C