Evaluate ∫ (e^(log x) + sin x) cos x dx

MathematicsIntegration by PartsJEE Advanced 1981Easy

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

Visualized Solution (Hindi)

Simplify .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } e^{l o g x}

Identify the term $e^{l o g x}$ inside the integral.
Apply the logarithmic identity: $e^{l o g a} = a$ .
The expression simplifies to: $x + sin x$ .

Distribute .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } cos x

Multiply the simplified expression by $cos x$ .
Apply the distributive property: $a (b + c) = ab + a c$ .
The integrand becomes: $x cos x + sin x cos x$ .

Split the Integral

Use the linearity property: $\int (f (x) + g (x)) d x = \int f (x) d x + \int g (x) d x$ .
Rewrite the problem as two separate integrals: $I = \int x cos x d x + \int sin x cos x d x$ .

Setup Integration by Parts

Focus on the first integral: $\int x cos x d x$ .
Apply the ILATE rule to choose $u$ and $d v$ .
Let $u = x$ (Algebraic) and $d v = cos x d x$ (Trigonometric).

Apply IBP Formula

Use the formula: $\int u d v = uv - \int v d u$ .
Substitute $u = x$ and $v = \int cos x d x = sin x$ .
The setup becomes: $x sin x - \int sin x d x$ .

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

Evaluate the remaining integral: $\int sin x d x = - cos x$ .
Substitute back: $x sin x - (- cos x)$ .
Result of the first part: $x sin x + cos x$ .

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

Focus on the second integral: $\int sin x cos x d x$ .
Apply the identity: $sin 2 x = 2 sin x cos x$ .
Rewrite the integral as: $\frac{1}{2} \int sin 2 x d x$ .

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

Integrate $sin 2 x$ using the rule: $\int sin (a x) d x = - \frac{c o s ( a x )}{a}$ .
The integral is: $\frac{1}{2} (- \frac{c o s 2 x}{2})$ .
Result of the second part: $- \frac{1}{4} cos 2 x$ .

Final Result and Takeaways

Combine the results from both parts.
Add the constant of integration $C$ .
Final Answer: $x sin x + cos x - \frac{1}{4} cos 2 x + C$ .

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Conceptually Similar Problems

MathematicsEvaluation of Special Integral FormsJEE Advanced 1978Easy

View in:English Hindi

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 1999Easy

View in:English Hindi

Integrate $\int_{0}^{π} \frac{e ^{c o s x}}{e ^{c o s x} + e ^{- c o s x}} d x$ .

Answer:

π /2

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

MathematicsFundamental Theorem & Properties of Definite IntegralsJEE Advanced 2005Moderate

View in:English Hindi

Evaluate $\int_{0}^{π} e^{∣ c o s x ∣} (2 sin (\frac{1}{2} cos x) + 3 cos (\frac{1}{2} cos x)) sin x d x$ .

Answer:

24 [e cos (\frac{1}{2}) + \frac{1}{2} e sin (\frac{1}{2}) - 1]

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

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}

MathematicsIntegration by PartsJEE Advanced 1994Moderate

View in:English Hindi

Find the indefinite integral $\int cos 2 θ ln (\frac{c o s θ + s i n θ}{c o s θ - s i n θ}) d θ$

Answer:

\frac{s i n 2 θ}{2} ln (\frac{c o s θ + s i n θ}{c o s θ - s i n θ}) - \frac{1}{2} ln sec 2 θ + C

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

MathematicsFundamental Theorem & Properties of Definite IntegralsJEE Advanced 1986Moderate

View in:English Hindi

Evaluate: $\int_{0}^{π} \frac{x d x}{1 + c o s α s i n x}, 0 < α < π$

Answer:

\frac{π α}{s i n α}

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

Visualized Solution (Hindi)

Simplify $e^{l o g x}$

Distribute $cos x$

Split the Integral

Setup Integration by Parts

Apply IBP Formula

Integrate $sin x$

Use $sin 2 x$ Identity

Integrate $sin 2 x$

Final Result and Takeaways

Conceptually Similar Problems

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

Integrate $\int_{0}^{π} \frac{e ^{c o s x}}{e ^{c o s x} + e ^{- c o s x}} d x$ .

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

Evaluate $\int_{0}^{π} e^{∣ c o s x ∣} (2 sin (\frac{1}{2} cos x) + 3 cos (\frac{1}{2} cos x)) sin x d x$ .

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

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

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

Find the indefinite integral $\int cos 2 θ ln (\frac{c o s θ + s i n θ}{c o s θ - s i n θ}) d θ$

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

Evaluate: $\int_{0}^{π} \frac{x d x}{1 + c o s α s i n x}, 0 < α < π$

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

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

Integrate $\int_{0}^{π} \frac{e ^{c o s x}}{e ^{c o s x} + e ^{- c o s x}} d x$ .

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

Evaluate $\int_{0}^{π} e^{∣ c o s x ∣} (2 sin (\frac{1}{2} cos x) + 3 cos (\frac{1}{2} cos x)) sin x d x$ .

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

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

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

Find the indefinite integral $\int cos 2 θ ln (\frac{c o s θ + s i n θ}{c o s θ - s i n θ}) d θ$

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

Evaluate: $\int_{0}^{π} \frac{x d x}{1 + c o s α s i n x}, 0 < α < π$

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate ∫(elogx+sinx)cosxdx

Visualized Solution (Hindi)

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

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

Split the Integral

Setup Integration by Parts

Apply IBP Formula

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

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

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

Final Result and Takeaways

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

Simplify $e^{l o g x}$

Distribute $cos x$

Integrate $sin x$

Use $sin 2 x$ Identity

Integrate $sin 2 x$

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

Integrate $\int_{0}^{π} \frac{e ^{c o s x}}{e ^{c o s x} + e ^{- c o s x}} d x$ .

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

Evaluate $\int_{0}^{π} e^{∣ c o s x ∣} (2 sin (\frac{1}{2} cos x) + 3 cos (\frac{1}{2} cos x)) sin x d x$ .

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

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

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

Find the indefinite integral $\int cos 2 θ ln (\frac{c o s θ + s i n θ}{c o s θ - s i n θ}) d θ$

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

Evaluate: $\int_{0}^{π} \frac{x d x}{1 + c o s α s i n x}, 0 < α < π$

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

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

Integrate $\int_{0}^{π} \frac{e ^{c o s x}}{e ^{c o s x} + e ^{- c o s x}} d x$ .

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

Evaluate $\int_{0}^{π} e^{∣ c o s x ∣} (2 sin (\frac{1}{2} cos x) + 3 cos (\frac{1}{2} cos x)) sin x d x$ .

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

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

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

Find the indefinite integral $\int cos 2 θ ln (\frac{c o s θ + s i n θ}{c o s θ - s i n θ}) d θ$

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

Evaluate: $\int_{0}^{π} \frac{x d x}{1 + c o s α s i n x}, 0 < α < π$