Evaluate ∫ (√tan x + √cot x) dx

MathematicsIntegration by SubstitutionJEE Advanced 1989Moderate

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

Answer:

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

Visualized Solution (English)

Rewrite in Terms of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } sin x and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } cos x

Evaluate $I = \int (tan x + cot x) d x$
Rewrite using $tan x = \frac{s i n x}{c o s x}$ and $cot x = \frac{c o s x}{s i n x}$ :
$I = \int (\frac{s i n x}{c o s x} + \frac{c o s x}{s i n x}) d x$

Combine Using Common Denominator

Take the LCM of the denominators:
$I = \int \frac{( s i n x \cdot s i n x ) + ( c o s x \cdot c o s x )}{s i n x c o s x} d x$
$I = \int \frac{s i n x + c o s x}{s i n x c o s x} d x$

The .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2 Manipulation

Multiply and divide by $2$ to create $sin 2 x$ in the denominator:
$I = 2 \int \frac{s i n x + c o s x}{2 s i n x c o s x} d x$
Using $2 sin x cos x = sin 2 x$ :
$I = 2 \int \frac{s i n x + c o s x}{s i n 2 x} d x$

Choosing the Right Substitution

Look for a function whose derivative is $(sin x + cos x)$ :
Let $t = sin x - cos x$
Differentiating both sides with respect to $x$ :
$\frac{d t}{d x} = cos x - (- sin x) = cos x + sin x$
So, $d t = (sin x + cos x) d x$

Relating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } sin 2 x to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } t

Square the substitution to find $sin 2 x$ :
$t^{2} = (sin x - cos x)^{2}$
$t^{2} = sin^{2} x + cos^{2} x - 2 sin x cos x$
Substitute $sin^{2} x + cos^{2} x = 1$ and $2 sin x cos x = sin 2 x$ :
$t^{2} = 1 - sin 2 x ⟹ sin 2 x = 1 - t^{2}$

Substitute and Integrate

Substitute $d t$ and $sin 2 x$ into the integral:
$I = 2 \int \frac{d t}{1 - t ^{2}}$
Apply the standard integral formula $\int \frac{1}{1 - x ^{2}} d x = sin^{- 1} x + C$ :
$I = 2 sin^{- 1} t + C$

Final Back-substitution

Substitute $t = sin x - cos x$ back into the result:
$I = 2 sin^{- 1} (sin x - cos x) + C$
Note: This is equivalent to the form $2 tan^{- 1} (\frac{t a n x - c o t x}{2}) + C$ as requested in some contexts.

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Conceptually Similar Problems

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

Answer:

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

MathematicsEvaluation of Special Integral FormsJEE Advanced 1978Easy

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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

MathematicsFundamental Theorem & Properties of Definite IntegralsJEE Advanced 1988Moderate

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

Answer:

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

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Evaluate $\int sin^{- 1} (\frac{2 x + 2}{4 x ^{2} + 8 x + 13}) d x$

Answer:

(x + 1) tan^{- 1} (\frac{2 x + 2}{3}) - \frac{3}{4} lo g (4 x^{2} + 8 x + 13) + C

MathematicsIntegration by SubstitutionJEE Advanced 1987Moderate

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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

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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

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$\int \frac{d x}{c o s x + 3 s i n x}$ equals

(A)

lo g tan (\frac{x}{2} + \frac{π}{12}) + C

(B)

lo g tan (\frac{x}{2} - \frac{π}{12}) + C

(C)

\frac{1}{2} lo g tan (\frac{x}{2} + \frac{π}{12}) + C

(D)

\frac{1}{2} lo g tan (\frac{x}{2} - \frac{π}{12}) + C

Answer:C

MathematicsIntegration by SubstitutionJEE Advanced 1996Moderate

View in:English Hindi

Evaluate $\int \frac{( x + 1 )}{x ( 1 + x e ^{x} ) ^{2}} d x$

Answer:

lo g (\frac{1 + x e ^{x}}{x e ^{x}}) - \frac{1}{1 + x e ^{x}} + C

MathematicsFundamental Theorem & Properties of Definite IntegralsJEE Advanced 1997Moderate

View in:English Hindi

Determine the value of $\int_{- π}^{π} \frac{2 x ( 1 + s i n x )}{1 + c o s ^{2} x} d x$ .

Answer:

π^{2}

MathematicsFundamental Theorem & Properties of Definite IntegralsJEE Advanced 2004Moderate

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Find the value of $\int_{- π /3}^{π /3} \frac{π + 4 x ^{3}}{2 - c o s ( ∣ x ∣ + \frac{π}{3} )} d x$ .

Answer:

\frac{4 π}{3} [tan^{- 1} 3 - \frac{π}{4}]

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

Visualized Solution (English)

Rewrite in Terms of $sin x$ and $cos x$

Combine Using Common Denominator

The $2$ Manipulation

Choosing the Right Substitution

Relating $sin 2 x$ to $t$

Substitute and Integrate

Final Back-substitution

Conceptually Similar Problems

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

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

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

Evaluate $\int sin^{- 1} (\frac{2 x + 2}{4 x ^{2} + 8 x + 13}) d x$

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

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

$\int \frac{d x}{c o s x + 3 s i n x}$ equals

Evaluate $\int \frac{( x + 1 )}{x ( 1 + x e ^{x} ) ^{2}} d x$

Determine the value of $\int_{- π}^{π} \frac{2 x ( 1 + s i n x )}{1 + c o s ^{2} x} d x$ .

Find the value of $\int_{- π /3}^{π /3} \frac{π + 4 x ^{3}}{2 - c o s ( ∣ x ∣ + \frac{π}{3} )} d x$ .

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

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

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

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

Evaluate $\int sin^{- 1} (\frac{2 x + 2}{4 x ^{2} + 8 x + 13}) d x$

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

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

$\int \frac{d x}{c o s x + 3 s i n x}$ equals

Evaluate $\int \frac{( x + 1 )}{x ( 1 + x e ^{x} ) ^{2}} d x$

Determine the value of $\int_{- π}^{π} \frac{2 x ( 1 + s i n x )}{1 + c o s ^{2} x} d x$ .

Find the value of $\int_{- π /3}^{π /3} \frac{π + 4 x ^{3}}{2 - c o s ( ∣ x ∣ + \frac{π}{3} )} d x$ .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate ∫(tanx​+cotx​)dx

Visualized Solution (English)

Rewrite in Terms of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } sinx and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } cosx

Combine Using Common Denominator

The .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2​ Manipulation

Choosing the Right Substitution

Relating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } sin2x to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } t

Substitute and Integrate

Final Back-substitution

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The $2$ Manipulation

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