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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
Assertion (A) : The value of$\int_{-3}^3\left(a x^5+b x^3+c x+k\right) d x,$ where $a, b, c, k$ are constants, depends on only $k$.
Reason (R) : $\int_{-a}^a f(x) d x=0$, if $f(-x)=-f(x)$ i.e., $f$ is an odd function.
  • ✓
    Both (A) and (R) are true and (R) is the correct explanation of (A).
  • B
    Both (A) and (R) are true but (R) is not the correct explanation of (A).
  • C
    (A) is true but (R) is false.
  • D
    (A) is false but (R) is true.
✓

Answer

Correct option: A.
Both (A) and (R) are true and (R) is the correct explanation of (A).
(a) : Clearly, Reason is true.
Let $I=\int_{-3}^3\left(a x^5+b x^3+c x+k\right) d x$
$
=a \int_{-3}^3 x^5 d x+b \int_{-3}^3 x^3 d x+c \int_{-3}^3 x d x+k \int_{-3}^3 1 d x
$
Since, $x^5, x^3, x$ are odd function
$
\therefore \quad I=0+0+0+k[x]_{-3}^3=6 k \text {, }
$
which is dependent only on $k$.

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