Evaluate the following ∫ dx / (x²(x⁴ +...

MathematicsIntegration by SubstitutionJEE Advanced 1984Easy

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

Answer:

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

Visualized Solution (English)

Understanding the Structure of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{\frac{3}{4}}}

Given integral: $I = \int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{\frac{3}{4}}}$
Observe the denominator structure: a power of $x$ outside and a binomial with a higher power of $x$ inside a fractional power.

Factoring Out .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x^{4} from the Parentheses

Factor out $x^{4}$ from $(x^{4} + 1)^{\frac{3}{4}}$ :
$(x^{4} + 1)^{\frac{3}{4}} = [x^{4} (1 + \frac{1}{x ^{4}})]^{\frac{3}{4}}$

Simplifying the Denominator Powers

Apply the power rule $(a \cdot b)^{n} = a^{n} \cdot b^{n}$ :
$(x^{4})^{\frac{3}{4}} \cdot (1 + \frac{1}{x ^{4}})^{\frac{3}{4}} = x^{3} \cdot (1 + \frac{1}{x ^{4}})^{\frac{3}{4}}$
Combine with the existing $x^{2}$ : $x^{2} \cdot x^{3} = x^{5}$
The integral becomes: $I = \int \frac{d x}{x ^{5} ( 1 + \frac{1}{x ^{4}} ) ^{\frac{3}{4}}}$

Choosing the Substitution .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } t = 1 + \frac{1}{x ^{4}}

Let $t = 1 + \frac{1}{x ^{4}}$
This can be written as $t = 1 + x^{- 4}$ for easier differentiation.

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

Differentiate $t$ with respect to $x$ :
$\frac{d t}{d x} = 0 + (- 4) x^{- 5}$
$d t = - \frac{4}{x ^{5}} d x$

Isolating the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \frac{d x}{x ^{5}} Term

Rearrange to match the integral's numerator:
$\frac{d x}{x ^{5}} = - \frac{d t}{4}$

Substituting into the Integral

Substitute $t$ and $d t$ into the integral:
$I = \int \frac{1}{t ^{\frac{3}{4}}} \cdot (- \frac{d t}{4})$
$I = - \frac{1}{4} \int t^{- \frac{3}{4}} d t$

Integrating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } t^{- \frac{3}{4}} using Power Rule

Apply the power rule $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1}$ :
$I = - \frac{1}{4} [\frac{t ^{- \frac{3}{4} + 1}}{- \frac{3}{4} + 1}] + C$
$I = - \frac{1}{4} [\frac{t ^{\frac{1}{4}}}{\frac{1}{4}}] + C$

Simplifying the Result

Simplify the coefficients:
$I = - \frac{1}{4} \cdot 4 \cdot t^{\frac{1}{4}} + C$
$I = - t^{\frac{1}{4}} + C$

Final Back-substitution and Conclusion

Substitute $t = 1 + \frac{1}{x ^{4}}$ back:
$I = - (1 + \frac{1}{x ^{4}})^{\frac{1}{4}} + C$
Key Takeaway: Factoring out the highest power from a binomial inside a fractional power often creates a perfect substitution form.
Next Challenge: Try evaluating $\int \frac{d x}{x ^{3} ( x ^{6} + 1 ) ^{\frac{2}{3}}}$ using the same method.

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Conceptually Similar Problems

MathematicsIntegration by SubstitutionJEE Main 2015Easy

View in:English Hindi

The integral $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$ equals

(A)

- (x^{4} + 1)^{1/4} + c

(B)

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

(C)

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

(D)

(x^{4} + 1)^{1/4} + c

Answer:B

MathematicsEvaluation of Special Integral FormsJEE Advanced 1983Easy

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

Answer:

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

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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

MathematicsFundamental Theorem & Properties of Definite IntegralsJEE Advanced 1995Moderate

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Evaluate the definite integral : $\int_{- 1/ 3}^{1/ 3} (\frac{x ^{4}}{1 - x ^{4}}) cos^{- 1} (\frac{2 x}{1 + x ^{2}}) d x$

Answer:

\frac{π}{12} [π + 3 lo g_{e} (2 + 3) - 43]

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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

MathematicsFundamental Theorem & Properties of Definite IntegralsJEE Advanced 1993Moderate

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Evaluate $\int_{2}^{3} \frac{2 x ^{5} + x ^{4} - 2 x ^{3} + 2 x ^{2} + 1}{( x ^{2} + 1 ) ( x ^{4} - 1 )} d x$

Answer:

\frac{1}{2} lo g 6 - \frac{1}{10}

MathematicsIntegration by SubstitutionJEE Advanced 2006Moderate

View in:English Hindi

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

(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

MathematicsIntegration by SubstitutionJEE Advanced 1985Moderate

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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

MathematicsIntegration by SubstitutionJEE Advanced 1979Easy

View in:English Hindi

Evaluate $\int \frac{x ^{2} d x}{( a + b x ) ^{2}}$

Answer:

\frac{1}{b ^{3}} [a + b x - 2 a lo g ∣ a + b x ∣ - \frac{a ^{2}}{a + b x}] + C

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

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

Visualized Solution (English)

Understanding the Structure of $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{\frac{3}{4}}}$

Factoring Out $x^{4}$ from the Parentheses

Simplifying the Denominator Powers

Choosing the Substitution $t = 1 + \frac{1}{x ^{4}}$

Differentiating to Find $d t$

Isolating the $\frac{d x}{x ^{5}}$ Term

Substituting into the Integral

Integrating $t^{- \frac{3}{4}}$ using Power Rule

Simplifying the Result

Final Back-substitution and Conclusion

Conceptually Similar Problems

The integral $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$ equals

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

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

The integral $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$ equals

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Evaluate $\int_{2}^{3} \frac{2 x ^{5} + x ^{4} - 2 x ^{3} + 2 x ^{2} + 1}{( x ^{2} + 1 ) ( x ^{4} - 1 )} d x$

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate the following ∫x2(x4+1)3/4dx​

Visualized Solution (English)

Understanding the Structure of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∫x2(x4+1)43​dx​

Factoring Out .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x4 from the Parentheses

Simplifying the Denominator Powers

Choosing the Substitution .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } t=1+x41​

Differentiating to Find .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } dt

Isolating the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x5dx​ Term

Substituting into the Integral

Integrating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } t−43​ using Power Rule

Simplifying the Result

Final Back-substitution and Conclusion

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

Understanding the Structure of $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{\frac{3}{4}}}$

Factoring Out $x^{4}$ from the Parentheses

Choosing the Substitution $t = 1 + \frac{1}{x ^{4}}$

Differentiating to Find $d t$

Isolating the $\frac{d x}{x ^{5}}$ Term

Integrating $t^{- \frac{3}{4}}$ using Power Rule

The integral $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$ equals

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

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

Evaluate the definite integral : $\int_{- 1/ 3}^{1/ 3} (\frac{x ^{4}}{1 - x ^{4}}) cos^{- 1} (\frac{2 x}{1 + x ^{2}}) d x$

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

Evaluate $\int_{2}^{3} \frac{2 x ^{5} + x ^{4} - 2 x ^{3} + 2 x ^{2} + 1}{( x ^{2} + 1 ) ( x ^{4} - 1 )} d x$

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

The integral $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$ equals

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

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

Evaluate the definite integral : $\int_{- 1/ 3}^{1/ 3} (\frac{x ^{4}}{1 - x ^{4}}) cos^{- 1} (\frac{2 x}{1 + x ^{2}}) d x$

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

Evaluate $\int_{2}^{3} \frac{2 x ^{5} + x ^{4} - 2 x ^{3} + 2 x ^{2} + 1}{( x ^{2} + 1 ) ( x ^{4} - 1 )} d x$

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

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

Evaluate $\int \frac{x ^{2} d x}{( a + b x ) ^{2}}$

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