_abf(x),dx=F(b)-F(a) when F =f and the integrand is continuous on the interval. The arbitrary constant cancels in the subtraction. Splitting at an intermediate point cancels the two copies of its antiderivative value. Find an antiderivative; substitute the upper limit first; subtract the lower-limit value. Reverse limits by negating the whole integral; add adjacent interval contributions. _03f(x),dx=7, _01f(x),dx=2 _31f(x),dx=-(7-2)=-5. Exam check: equal limits give zero. An integral may be negative; a definite integral does not need +C. Check undefined points first.

\(\int_a^bf(x)\,\mathrm{d}x=F(b)-F(a)\) when \(F^{\prime}=f\) and the integrand is continuous on the interval.

The arbitrary constant cancels in the subtraction. Splitting at an intermediate point cancels the two copies of its antiderivative value.

Find an antiderivative; substitute the upper limit first; subtract the lower-limit value. Reverse limits by negating the whole integral; add adjacent interval contributions.

\(\int_0^3f(x)\,\mathrm{d}x=7,\quad\int_0^1f(x)\,\mathrm{d}x=2\quad\Rightarrow\quad\int_3^1f(x)\,\mathrm{d}x=-(7-2)=-5\).

Exam check: equal limits give zero. An integral may be negative; a definite integral does not need \(+C\). Check undefined points first.

Integral limits and interval pieces

Integral limits and interval pieces