Fundamental Integration Formulas

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If $\int \frac{s i n x}{s i n ( x - α )} d x = A x + B lo g sin (x - α) + C$ , then value of $(A, B)$ is

(- cos α, sin α)

(cos α, sin α)

(- sin α, cos α)

(sin α, cos α)

JEE Advanced 2007

Let $F (x)$ be an indefinite integral of $sin^{2} x$ . \\ STATEMENT-1: The function $F (x)$ satisfies $F (x + π) = F (x)$ because \\ STATEMENT-2: $sin^{2} (x + π) = sin^{2} x$ for all real $x$ .

Statement-1 is True, statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Statement-1 is True, statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

Statement-1 is True, Statement-2 is False

Statement-1 is False, Statement-2 is True.

JEE Main 2019 (8 April Shift 1)

$\int \frac{s i n ( 5 x /2 )}{s i n ( x /2 )} d x$ is equal to : (where c is a constant of integration)

2 x + sin x + 2 sin 2 x + c

x + 2 sin x + 2 sin 2 x + c

x + 2 sin x + sin 2 x + c

2 x + sin x + sin 2 x + c

JEE Main 2020 (9 January Shift 1)

If $f^{'} (x) = tan^{- 1} (sec x + tan x), - \frac{π}{2} < x < \frac{π}{2}$ , and $f (0) = 0$ , then $f (1)$ is equal to:

\frac{π + 1}{4}

\frac{π + 2}{4}

\frac{1}{4}

\frac{π - 1}{4}