Question:
The value of $\displaystyle \int_{0}^{1} x(1-x)^4 \, dx$ is equal to:
The value of $\displaystyle \int_{0}^{1} x(1-x)^4 \, dx$ is equal to:
Show Hint
Expand polynomial first for easy definite integration.
Updated On: Apr 24, 2026
- $\frac{1}{60}$
- $\frac{1}{15}$
- $\frac{1}{30}$
- $\frac{1}{45}$
- $\frac{1}{20}$
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Concept:
• Expand and integrate term-wise
Step 1: Expand $(1-x)^4$
\[ (1-x)^4 = 1 - 4x + 6x^2 - 4x^3 + x^4 \]
Step 2: Multiply by $x$
\[ x(1-x)^4 = x - 4x^2 + 6x^3 - 4x^4 + x^5 \]
Step 3: Integrate term-wise
\[ \int_{0}^{1} x\,dx = \frac{1}{2} \] \[ \int_{0}^{1} x^2 dx = \frac{1}{3}, \quad \int x^3 dx = \frac{1}{4} \] \[ \int x^4 dx = \frac{1}{5}, \quad \int x^5 dx = \frac{1}{6} \]
Step 4: Combine
\[ = \frac{1}{2} - 4\cdot\frac{1}{3} + 6\cdot\frac{1}{4} - 4\cdot\frac{1}{5} + \frac{1}{6} \] \[ = \frac{1}{2} - \frac{4}{3} + \frac{3}{2} - \frac{4}{5} + \frac{1}{6} \] \[ = \frac{1}{30} \] Final Conclusion:
\[ = \frac{1}{30} \]
• Expand and integrate term-wise
Step 1: Expand $(1-x)^4$
\[ (1-x)^4 = 1 - 4x + 6x^2 - 4x^3 + x^4 \]
Step 2: Multiply by $x$
\[ x(1-x)^4 = x - 4x^2 + 6x^3 - 4x^4 + x^5 \]
Step 3: Integrate term-wise
\[ \int_{0}^{1} x\,dx = \frac{1}{2} \] \[ \int_{0}^{1} x^2 dx = \frac{1}{3}, \quad \int x^3 dx = \frac{1}{4} \] \[ \int x^4 dx = \frac{1}{5}, \quad \int x^5 dx = \frac{1}{6} \]
Step 4: Combine
\[ = \frac{1}{2} - 4\cdot\frac{1}{3} + 6\cdot\frac{1}{4} - 4\cdot\frac{1}{5} + \frac{1}{6} \] \[ = \frac{1}{2} - \frac{4}{3} + \frac{3}{2} - \frac{4}{5} + \frac{1}{6} \] \[ = \frac{1}{30} \] Final Conclusion:
\[ = \frac{1}{30} \]
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Definite Integral Questions
- $\int \frac{\left(\sin x + \cos x\right)\left(2 - \sin 2x\right)}{\sin^{2} 2x}dx = $
- If \(f(x) = \begin{cases} \cos x & for\ x\geq0 \\ 2x &for\ x\lt0 \end{cases}\), then the value of \(\displaystyle\int^{\frac{\pi}{2}}_{-2}dx\) is equal to
The area bounded by the parabola \( y = x^2 + 4 \) and the straight line passing through the points \( (-1,2) \quad {and} \quad (1,6) \) is (in square units)
- The value of \( \int_{-2}^{2} x |x| \,dx \) is:
- Evaluate the integral: \[ \int_{0}^{3} |x - 2| \,dx \] is equal to
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions