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CBSE Class 12 · 2023 · Definite IntegralsShort Answer

Evaluate

\int_{- 1}^{1} x^{4} - x d x

Answer: $1$

Solution

Factor

x^{4} - x = x (x - 1) (x^{2} + x + 1)

; the last factor is always positive, so the sign comes from

x (x - 1)

.
On

[- 1, 0]

:

x \leq 0

and

x - 1 < 0

, so

x^{4} - x \geq 0

and

∣ x^{4} - x ∣ = x^{4} - x

.
On

[0, 1]

:

x \geq 0

and

x - 1 \leq 0

, so

x^{4} - x \leq 0

and

∣ x^{4} - x ∣ = x - x^{4}

.

\int_{- 1}^{0} (x^{4} - x) d x = [\frac{x ^{5}}{5} - \frac{x ^{2}}{2}]_{- 1}^{0} = 0 - (- \frac{1}{5} - \frac{1}{2}) = \frac{7}{10}

\int_{0}^{1} (x - x^{4}) d x = [\frac{x ^{2}}{2} - \frac{x ^{5}}{5}]_{0}^{1} = \frac{1}{2} - \frac{1}{5} = \frac{3}{10}

I = \frac{7}{10} + \frac{3}{10} = 1

CBSE Class 12 Board · 2023 · Subjective

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