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UPSC NDA Math Model Paper 2

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Question : 74 of 120

Marks: +1, -0

The value of

\int\limits_{-2}^{2}(a x^{5}+b x^{3}+c)\,dx

depends on the value of:

b.
c.
a.
a and b.

Solution:

We observe that

ax^{5}

and

bx^{3}

are odd functions of

x

, because

a(-x)^{5} = -ax^{5}

and

b(-x)^{3} = -bx^{3}

.

\therefore \int\limits_{-2}^{2} a x^{5}\,dx = 0

and

\int\limits_{-2}^{2} b x^{3}\,dx = 0

The value of

\int\limits_{-2}^{2}(a x^{5}+b x^{3}+c)\,dx

depends on the value of

c

.

In fact,

\int\limits_{-2}^{2}(a x^{5}+b x^{3}+c)\,dx \int\limits_{-2}^{2}c\,dx = c[x]_{-2}^{2} = c[2-(-2)] = 4c

=c[x]_{-2}^{2}=c[2-(-2)]=4 c

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