Fresh features from the #1 AI-enhanced learning platform.Try it free
Fresh features from the #1 AI-enhanced learning platformCrush your year with the magic of personalized studying.Try it free
Question

# Let $I_{m,n} = \int_{0}^{1}x^{m}(1-x)^{n}\ dx$ for constant $m,n.$ Show that $I_{m,n} = I_{n,m}.$

Solution

Verified
Step 1
1 of 2

Using substitution, we have:

\begin{align*} \int_{0}^{1} x^m(1-x)^n\,dx&= % matrica supstitucije \color{#4257b2} \left[ \small { \begin{array} {rlcrl} 1-x&=t & \Rightarrow & -dx&=dt \\ x&=t-1 \\ x &= 0 & \Rightarrow & t&=1 \\ x &=1 & \Rightarrow & t&=0 \end{array} } \right] \color {black} = \\[15pt] &=-\int_{1}^{0} (1-t)^m\,t^n\,dt=\int _{0}^{1} (1-t)^m \,t^n\,dt= \\[21pt] &=\int _{0}^{1} t^n\,(1-t)^m\,dt=I_{n,m} \end{align*}

$\quad \quad \quad$ So,

$I_{m,n}=I_{n,m}$

## Recommended textbook solutions

#### Thomas' Calculus

14th EditionISBN: 9780134438986 (5 more)Christopher E Heil, Joel R. Hass, Maurice D. Weir
10,144 solutions

#### Calculus: Single and Multivariable

6th EditionISBN: 9780470888612 (1 more)Andrew M. Gleason, Deborah Hughes-Hallett, William G. McCallum
11,813 solutions

#### Calculus: Early Transcendentals

8th EditionISBN: 9781285741550 (5 more)James Stewart
11,085 solutions

#### Calculus: Early Transcendentals

9th EditionISBN: 9781337613927 (2 more)Daniel K. Clegg, James Stewart, Saleem Watson
11,050 solutions