Evaluate ∫ sin x / (sin x - cos x) dx

MathematicsEvaluation of Special Integral FormsJEE Advanced 1978Easy

Evaluate $\int \frac{s i n x}{s i n x - c o s x} d x$

Answer:

\frac{1}{2} lo g ∣ sin x - cos x ∣ + \frac{x}{2} + C

Visualized Solution (Hindi)

Understanding the Problem

Given Integral: $I = \int \frac{s i n x}{s i n x - c o s x} d x$
Our goal is to express the numerator in terms of the denominator and its derivative.
Denominator ( $D$ ): $sin x - cos x$
Derivative of Denominator ( $D^{'}$ ): $cos x + sin x$

The Strategic Manipulation

Multiply and divide by $2$ to manipulate the numerator:
$I = \frac{1}{2} \int \frac{2 s i n x}{s i n x - c o s x} d x$

Introducing the Denominator and Derivative

Rewrite $2 sin x$ as $(sin x + sin x)$ and add/subtract $cos x$ :
$2 sin x = (sin x - cos x) + (sin x + cos x)$
Substitute this back into the integral:
$I = \frac{1}{2} \int \frac{( s i n x - c o s x ) + ( s i n x + c o s x )}{s i n x - c o s x} d x$

Splitting the Integral

Split the fraction into two parts:
$I = \frac{1}{2} \int (\frac{s i n x - c o s x}{s i n x - c o s x} + \frac{s i n x + c o s x}{s i n x - c o s x}) d x$
Simplify the first term:
$I = \frac{1}{2} \int (1 + \frac{s i n x + c o s x}{s i n x - c o s x}) d x$

Integrating the Terms

Integrate term by term:
$\int 1 d x = x$
$\int \frac{s i n x + c o s x}{s i n x - c o s x} d x = lo g ∣ sin x - cos x ∣$
Using the property: $\int \frac{f ^{'} ( x )}{f ( x )} d x = lo g ∣ f (x) ∣ + C$

The Final Answer

Combine the results:
$I = \frac{1}{2} [x + lo g ∣ sin x - cos x ∣] + C$
Final simplified form:
$I = \frac{x}{2} + \frac{1}{2} lo g ∣ sin x - cos x ∣ + C$

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Conceptually Similar Problems

MathematicsIntegration by PartsJEE Advanced 1981Easy

View in:English Hindi

Evaluate $\int (e^{l o g x} + sin x) cos x d x$

Answer:

x sin x + cos x - \frac{1}{4} cos 2 x + C

MathematicsEvaluation of Special Integral FormsJEE Main 2004Moderate

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

(A)

\frac{1}{2} lo g tan (\frac{x}{2} + \frac{3 π}{8}) + C

(B)

\frac{1}{2} lo g cot (\frac{x}{2}) + C

(C)

\frac{1}{2} lo g tan (\frac{x}{2} - \frac{3 π}{8}) + C

(D)

\frac{1}{2} lo g tan (\frac{x}{2} - \frac{π}{8}) + C

Answer:A

MathematicsIntegration by SubstitutionJEE Advanced 1987Moderate

View in:English Hindi

Evaluate $\int \frac{( c o s 2 x ) ^{1/2}}{s i n x} d x$

Answer:

\frac{1}{2} lo g \frac{2 + 1 - t a n ^{2} x}{2 - 1 - t a n ^{2} x} - lo g (cot x + cot^{2} x - 1) + C

MathematicsIntegration by SubstitutionJEE Advanced 1985Moderate

View in:English Hindi

Evaluate the following $\int \frac{1 - x}{1 + x} d x$

Answer:

- 2 1 - x + cos^{- 1} x + x 1 - x + C

MathematicsFundamental Theorem & Properties of Definite IntegralsJEE Advanced 1985Moderate

View in:English Hindi

Evaluate the following: $\int_{0}^{π /2} \frac{x s i n x c o s x}{c o s ^{4} x + s i n ^{4} x} d x$

Answer:

\frac{π ^{2}}{16}

MathematicsFundamental Theorem & Properties of Definite IntegralsJEE Advanced 1984Easy

View in:English Hindi

Evaluate the following $\int_{0}^{1} \frac{x s i n ^{- 1} x}{1 - x ^{2}} d x$ .

Answer:1

MathematicsFundamental Theorem & Properties of Definite IntegralsJEE Advanced 1991Moderate

View in:English Hindi

Evaluate $\int_{0}^{π} \frac{x s i n 2 x s i n ( \frac{π}{2} c o s x )}{2 x - π} d x$

Answer:

\frac{8}{π ^{2}}

MathematicsIntegration by SubstitutionJEE Advanced 1989Moderate

View in:English Hindi

Evaluate $\int (tan x + cot x) d x$

Answer:

2 tan^{- 1} (\frac{t a n x - c o t x}{2}) + c

MathematicsIntegration by SubstitutionJEE Main 2008Easy

View in:English Hindi

The value of $2 \int \frac{s i n x d x}{s i n ( x - \frac{π}{4} )}$ is

(A)

x + lo g cos (x - \frac{π}{4}) + c

(B)

x - lo g sin (x - \frac{π}{4}) + c

(C)

x + lo g sin (x - \frac{π}{4}) + c

(D)

x - lo g cos (x - \frac{π}{4}) + c

Answer:C

MathematicsFundamental Theorem & Properties of Definite IntegralsJEE Advanced 1988Moderate

View in:English Hindi

Evaluate $\int_{0}^{1} lo g [1 - x + 1 + x] d x$

Answer:

\frac{1}{2} [lo g 2 + \frac{π}{2} - 1]

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate ∫sinx−cosxsinx​dx

Visualized Solution (Hindi)

Understanding the Problem

The Strategic Manipulation

Introducing the Denominator and Derivative

Splitting the Integral

Integrating the Terms

The Final Answer

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