Using ∫₀^a f(x) dx = ∫₀^a f(a - x) dx, evaluate ∫₀^1 x(1 - x)^5 dx.

Definite Integrals & Area · Question 50 of 71

Question 50

HardOriginal question

Using $\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx$ , evaluate $\int_0^1 x(1 - x)^5\,dx$ .

\frac{1}{42}

Correct answerYour answer

\frac{1}{30}

Correct answerYour answer

\frac{1}{7}

Correct answerYour answer

\frac{1}{6}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

Replacing $x$ by $1 - x$ : $\int_0^1 (1 - x)x^5\,dx = \int_0^1 (x^5 - x^6)\,dx$ .
$= \frac{1}{6} - \frac{1}{7} = \frac{1}{42}$ .

Worth noting

$\frac{1}{6}$ integrates only the $x^5$ term and drops $-x^6$ .

Ten questions from Definite Integrals & Area, at your own pace.

Ten questions from Definite Integrals & Area, at your own pace.