I = x/x - x dx is equal to: - Mathematics MCQ Mathematics Mock Test - 10 2026 | ExamTest.live

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I = \int \frac{s i n x}{s i n x - c o s x} d x is equal to:

Solution & Step-by-step Explanation

I = \int \frac{s i n x}{s i n x - c o s x} d x Multiply and divide by 2: I = \frac{1}{2} \int \frac{2 s i n x}{s i n x - c o s x} d x Rewrite the numerator: 2 sin x = (sin x - cos x) + (sin x + cos x) I = \frac{1}{2} \int [\frac{s i n x - c o s x}{s i n x - c o s x} + \frac{s i n x + c o s x}{s i n x - c o s x}] d x I = \frac{1}{2} \int 1 d x + \frac{1}{2} \int \frac{s i n x + c o s x}{s i n x - c o s x} d x For the second integral, let t = sin x - cos x, then d t = (cos x + sin x) d x . I = \frac{x}{2} + \frac{1}{2} \int \frac{1}{t} d t I = \frac{x}{2} + \frac{1}{2} ln ∣ t ∣ + c I = \frac{x}{2} + \frac{1}{2} ln ∣ sin x - cos x ∣ + c

Try it yourself before checking the explanation above.

I = \int \frac{s i n x}{s i n x - c o s x} d x is equal to:

\frac{x}{2} - \frac{1}{2} ln ∣ sin x - cos x ∣ + c

\frac{x}{2} + \frac{1}{2} ln ∣ sin x - cos x ∣ + c

\frac{x}{2} + \frac{1}{2} ln ∣ sin x + cos x ∣ + c

\frac{x}{2} - \frac{1}{2} ln ∣ sin x + cos x ∣ + c

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