MHT CET · Maths · Definite Integration
\(\int_{0}^{1} x(1-x)^{5} d x=\)
- A \(\frac{1}{7}\)
- B \(-\frac{1}{42}\)
- C \(\frac{1}{42}\)
- D \(\frac{1}{6}\)
Answer & Solution
Correct Answer
(C) \(\frac{1}{42}\)
Step-by-step Solution
Detailed explanation
(C)
\(\text {Let } I =\int_{0}^{1} x(1-x)^{5} d x \)
\( \therefore I =\int_{0}^{1}(1-x)(1-(1-x))^{5} d x=\int_{0}^{1}(1-x) x^{5}\) \(d x=\int\left(x^{5}-x^{6}\right) d x \)
\( =\left[\frac{x^{6}}{6}-\frac{x^{7}}{7}\right]_{0}^{1}=\frac{1}{6}-\frac{1}{7}=\frac{1}{42}\)
\(\text {Let } I =\int_{0}^{1} x(1-x)^{5} d x \)
\( \therefore I =\int_{0}^{1}(1-x)(1-(1-x))^{5} d x=\int_{0}^{1}(1-x) x^{5}\) \(d x=\int\left(x^{5}-x^{6}\right) d x \)
\( =\left[\frac{x^{6}}{6}-\frac{x^{7}}{7}\right]_{0}^{1}=\frac{1}{6}-\frac{1}{7}=\frac{1}{42}\)
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