Definite Integrals of Even and Odd Functions | NIMCET

Quick answer: For an even function ($f(-x) = f(x)$), $\int_{-a}^{a}f(x)\,dx = 2\int_0^a f(x)\,dx$. For an odd function ($f(-x) = -f(x)$), $\int_{-a}^{a}f(x)\,dx = 0$. For a periodic function with period T, $\int_0^{nT}f(x)\,dx = n\int_0^T f(x)\,dx$. The Leibniz rule gives $\frac{d}{dx}\int_{g(x)}^{h(x)}f(t)\,dt = f(h(x))\,h'(x) - f(g(x))\,g'(x)$.

Syllabus note: Reports on the revised NIMCET syllabus list definite integrals of even and odd functions, while the Leibniz rule and periodic functions are reported as removed. Check the official information brochure for your exam year; see our revised syllabus update.

This continues from definite integrals and King's property.

Even and odd functions

Any function splits as $f(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2}$ (even part + odd part). Only the even part survives in $\int_{-a}^{a}$.

Symmetric-limit results

Periodic functions

If $f(x + T) = f(x)$:

Common periods: $\sin x$ and $\cos x$ have period $2\pi$; $\lvert\sin x\rvert$, $\tan x$ and $\sin^2 x$ have period π; $\{x\} = x - [x]$ has period 1.

Leibniz rule

$\frac{d}{dx}\int_{g(x)}^{h(x)}f(t)\,dt = f(h(x))\,h'(x) - f(g(x))\,g'(x)$

Worked examples

Example 1: an odd integrand

$\int_{-1}^{1}x^3\cos x\,dx = 0$

since $x^3\cos x$ is odd.

Example 2: an even integrand

$\int_{-2}^{2}(x^2 + \lvert x\rvert)\,dx = 2\int_0^2(x^2 + x)\,dx = 2\left(\frac{8}{3} + 2\right) = \frac{28}{3}$

Example 3: mixed

Evaluate $\int_{-1}^{1}(x^5 + 3x^2 + \sin x)\,dx$.

The odd parts $x^5$ and $\sin x$ vanish, leaving $2\int_0^1 3x^2\,dx = 2$.

Example 4: a periodic function

Evaluate $\int_0^{10\pi}\lvert\sin x\rvert\,dx$.

The period is π and $\int_0^{\pi}\sin x\,dx = 2$, so the answer is $10 \times 2 = 20$.

Example 5: fractional part

Evaluate $\int_0^5\{x\}\,dx$.

$\int_0^1 x\,dx = \frac{1}{2}$, so the answer is $5 \times \frac{1}{2} = \frac{5}{2}$.

Example 6: Leibniz rule

Find $\frac{d}{dx}\int_x^{x^2}\frac{dt}{t}$ for $x > 0$.

$\frac{1}{x^2} \cdot 2x - \frac{1}{x} \cdot 1 = \frac{2}{x} - \frac{1}{x} = \frac{1}{x}$

Summary table

Exam tips

• With symmetric limits, check parity before doing any integration.
• Split mixed integrands into odd and even parts.
• For long intervals of periodic functions, find one period's value and multiply.

See the area under curves for geometric applications, and the NIMCET Calculus chapter.

Common mistakes

• Using the odd-function result when limits are not symmetric.
• Treating $\lvert x\rvert$ or $x\sin x$ as odd.
• Taking $2\pi$ as the period of $\lvert\sin x\rvert$.
• Forgetting the lower-limit term in the Leibniz rule.

Frequently Asked Questions

What is the integral of an odd function from −a to a?

It is 0.

What is the integral of an even function from −a to a?

It is 2∫₀ᵃ f(x) dx.

How do you integrate a periodic function over many periods?

Multiply the integral over one period by the number of periods.

What is the Leibniz rule?

d/dx ∫ from g(x) to h(x) of f(t) dt = f(h(x))h′(x) − f(g(x))g′(x).

Is the Leibniz rule in the NIMCET syllabus?

Reports on the revised syllabus list it as removed. Confirm with the official information brochure for your exam year.