Evaluate ∫ (x - 1)e^x / (x + 1)³ dx

MathematicsEvaluation of Special Integral FormsJEE Advanced 1983Easy

Evaluate $\int \frac{( x - 1 ) e ^{x}}{( x + 1 ) ^{3}} d x$

Answer:

\frac{e ^{x}}{( x + 1 ) ^{2}} + C

Visualized Solution (Hindi)

The Integral Challenge

Evaluate $I = \int \frac{( x - 1 ) e ^{x}}{( x + 1 ) ^{3}} d x$
Observe the presence of $e^{x}$ multiplied by a rational function.
Goal: Transform the integrand into a recognizable shortcut form.

The .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } e^{x} [f (x) + f^{'} (x)] Rule

Recall the identity: $\int e^{x} [f (x) + f^{'} (x)] d x = e^{x} f (x) + C$
This identity simplifies integration significantly when the derivative relationship exists.
We need to manipulate $\frac{x - 1}{( x + 1 ) ^{3}}$ to match $f (x) + f^{'} (x)$ .

Adjusting the Numerator

Rewrite the numerator $(x - 1)$ in terms of $(x + 1)$ .
$x - 1 = (x + 1) - 2$
Substitute this back into the integral: $I = \int e^{x} [\frac{( x + 1 ) - 2}{( x + 1 ) ^{3}}] d x$

Splitting the Fraction

Distribute the denominator: $I = \int e^{x} [\frac{x + 1}{( x + 1 ) ^{3}} - \frac{2}{( x + 1 ) ^{3}}] d x$
Simplify the first fraction: $\frac{x + 1}{( x + 1 ) ^{3}} = \frac{1}{( x + 1 ) ^{2}}$
The integral becomes: $I = \int e^{x} [\frac{1}{( x + 1 ) ^{2}} - \frac{2}{( x + 1 ) ^{3}}] d x$

Identifying .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f (x) and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f^{'} (x)

Let $f (x) = \frac{1}{( x + 1 ) ^{2}} = (x + 1)^{- 2}$
Differentiate using the power rule: $f^{'} (x) = - 2 (x + 1)^{- 3} = - \frac{2}{( x + 1 ) ^{3}}$
The integrand is exactly in the form $e^{x} [f (x) + f^{'} (x)]$ , where $f^{'} (x) = - \frac{2}{( x + 1 ) ^{3}}$ .

Final Result

Using the identity $\int e^{x} [f (x) + f^{'} (x)] d x = e^{x} f (x) + C$ :
$I = e^{x} \cdot \frac{1}{( x + 1 ) ^{2}} + C$
Final Answer: $I = \frac{e ^{x}}{( x + 1 ) ^{2}} + C$

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Conceptually Similar Problems

MathematicsIntegration by SubstitutionJEE Advanced 1996Moderate

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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 1984Easy

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Evaluate the following $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$

Answer:

- (1 + \frac{1}{x ^{4}})^{1/4} + C

MathematicsIntegration using Partial FractionsJEE Advanced 1999Moderate

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Integrate $\int \frac{x ^{3} + 3 x + 2}{( x ^{2} + 1 ) ^{2} ( x + 1 )} d x$

Answer:

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

MathematicsIntegration by PartsJEE Main 2014Moderate

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The integral $\int (1 + x - \frac{1}{x}) e^{x + 1/ x} d x$ is equal to

(A)

(x + 1) e^{x + 1/ x} + c

(B)

- x e^{x + 1/ x} + c

(C)

(x - 1) e^{x + 1/ x} + c

(D)

x e^{x + 1/ x} + c

Answer:D

MathematicsEvaluation of Special Integral FormsJEE Main 2005Easy

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$\int {\frac{l o g x - 1}{1 + ( l o g x ) ^{2}}}^{2} d x$ is equal to

(A)

\frac{l o g x}{( l o g x ) ^{2} + 1} + C

(B)

\frac{x}{x ^{2} + 1} + C

(C)

\frac{x e ^{x}}{1 + x ^{2}} + C

(D)

\frac{x}{( l o g x ) ^{2} + 1} + C

Answer:D

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

MathematicsIntegration by PartsJEE Advanced 1981Easy

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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

MathematicsIntegration by SubstitutionJEE Advanced 2006Moderate

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$\int \frac{x ^{2} - 1}{x ^{3} 2 x ^{4} - 2 x ^{2} + 1} d x =$

(A)

\frac{2 x ^{4} - 2 x ^{2} + 1}{x ^{2}} + c

(B)

\frac{2 x ^{4} - 2 x ^{2} + 1}{x ^{3}} + c

(C)

\frac{2 x ^{4} - 2 x ^{2} + 1}{x} + c

(D)

\frac{2 x ^{4} - 2 x ^{2} + 1}{2 x ^{2}} + c

Answer:D

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

MathematicsIntegration by SubstitutionJEE Main 2016Moderate

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The integral $\int \frac{2 x ^{12} + 5 x ^{9}}{( x ^{5} + x ^{3} + 1 ) ^{3}} d x$ is equal to

(A)

\frac{x ^{5}}{2 ( x ^{5} + x ^{3} + 1 ) ^{2}} + C

(B)

\frac{- x ^{10}}{2 ( x ^{5} + x ^{3} + 1 ) ^{2}} + C

(C)

\frac{- x ^{5}}{( x ^{5} + x ^{3} + 1 ) ^{2}} + C

(D)

\frac{x ^{10}}{2 ( x ^{5} + x ^{3} + 1 ) ^{2}} + C

Answer:D

Evaluate $\int \frac{( x - 1 ) e ^{x}}{( x + 1 ) ^{3}} d x$

Visualized Solution (Hindi)

The Integral Challenge

The $e^{x} [f (x) + f^{'} (x)]$ Rule

Adjusting the Numerator

Splitting the Fraction

Identifying $f (x)$ and $f^{'} (x)$

Final Result

Conceptually Similar Problems

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

Evaluate the following $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$

Integrate $\int \frac{x ^{3} + 3 x + 2}{( x ^{2} + 1 ) ^{2} ( x + 1 )} d x$

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Evaluate $\int \frac{( x - 1 ) e ^{x}}{( x + 1 ) ^{3}} d x$

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Evaluate the following $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$

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

The integral $\int \frac{2 x ^{12} + 5 x ^{9}}{( x ^{5} + x ^{3} + 1 ) ^{3}} d x$ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate ∫(x+1)3(x−1)ex​dx

Visualized Solution (Hindi)

The Integral Challenge

The .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ex[f(x)+f′(x)] Rule

Adjusting the Numerator

Splitting the Fraction

Identifying .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x) and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f′(x)

Final Result

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