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Definite integral methods

How to Evaluate Definite Integrals: Bounds and Worked Steps

To evaluate a definite integral, find an antiderivative and subtract its value at the lower bound from its value at the upper bound. Before doing that subtraction, check the domain, the bounds and what the question asks you to measure.

Direct answer

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,\mathrm{d}x=F(b)-F(a)

This form of the fundamental theorem applies when f is continuous on the closed finite interval and F′=f. Singularities and improper integrals need separate analysis.

How to evaluate a definite integral step by step

Read the lower bound a and upper bound b exactly as written. Check that the integrand is defined throughout the interval; a pole such as 1/x at zero cannot be handled by blindly subtracting endpoint values.

Find F with F′=f. A polynomial may use the power rule; a composite expression may need substitution; a product may need integration by parts. The method is chosen from the integrand, not from the presence of bounds.

Calculate F(b)−F(a), using parentheses around the complete lower-bound value. The same constant C cancels, so the definite result does not need +C.

If you substitute u=g(x), either transform both bounds and integrate in u, or return to x before using the original bounds. For ∫₀¹ 2x·exp(x²) dx, u=x² gives bounds 0 and 1 and the result e−1.

Check orientation and sign. Reversing bounds negates the result. An integral is a signed quantity; for total geometric area, split at sign changes and add nonnegative contributions.

1
F′(x)=f(x)F'(x)=f(x)
2
∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,\mathrm{d}x=F(b)-F(a)
3
∫baf(x) dx=−∫abf(x) dx\int_b^a f(x)\,\mathrm{d}x=-\int_a^b f(x)\,\mathrm{d}x

Worked examples

∫02x2 dx\int_{0}^{2} x^{2}\,\mathrm{d}x
83\frac{8}{3}

Use F(x)=x³/3 and compute 8/3−0. A decimal in the calculator is an approximation to this exact value.

∫202 x dx\int_{2}^{0} 2\,x\,\mathrm{d}x
−4-4

F(0)−F(2)=0−4. Do not reorder the bounds without changing the sign.

∫−11x dx\int_{-1}^{1} x\,\mathrm{d}x
00

The two signed contributions cancel. The total geometric area is 1, not 0.

Common mistakes

  • • Subtracting F(b) from F(a).
  • • Mixing x-bounds with a u-antiderivative.
  • • Leaving +C in a numerical definite result.
  • • Applying endpoint subtraction across a singularity.
  • • Taking an absolute value of the final result to claim total area.

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

These links use finite bounds. Symbolic and numerical methods have different support limits; the displayed verification label describes the actual check, not a guarantee for every integrand.

定積分計算機:x2x^{2}定積分計算機:2 x2\,x定積分計算機:xx
Open the calculator without a preset: 定積分計算機

References

Frequently asked questions

Why does a definite integral have no +C?+

Any common constant in the antiderivative cancels in [F(b)+C]−[F(a)+C].

Do I change the bounds during substitution?+

Yes if you finish the calculation in the new variable. Otherwise return to the original variable before using the original bounds.

Can a definite integral be negative?+

Yes. It measures signed accumulation and also depends on the orientation of the bounds, not only geometric area.

Conclusion

Evaluate definite integrals by checking the interval first, choosing an antiderivative method and computing upper minus lower. Keep variable changes, signs and the distinction between integral and area visible in your work.

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