NIMCET 2017 Question Paper with solutions

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Question : 114 of 120

Marks: +1, -0

Evaluate

∫↖{1}↙{0}

x

(1-x)^n

dx

${-1}/{(n+1)(n+2)}$
${1}/{(n+1)(n+2)}$
(n+1) (n+2)
(n-1) (n-2)

Solution:

I =

∫↖{1}↙{0}

s

(1-x)^n

dx
=

∫↖{1}↙{0}

(1 - x)

x^n

dx

[∫↖{a}↙{0}f(x)dx=∫↖{a}↙{0}f(a-x)dx]

=

∫↖{1}↙{0}(x^n-x^{n+1})

dx
=

[{x^{n+1}/{n+1}-x^{n+2}/{n+2} }]^1_0

=

1/{n+1}-1/{n+2}

=

1/{(n+1)(n+2)}

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