32. Second Fundamental Theorem of Calculus

Dated: 30-10-2024

Dummy Variable

If we change the letter for the variable of integration¹ but don't change the limits, then values of definite integrals² remains the same

\[\int_{1}^{2}x^{2}dx=\frac{x^{3}}{3}\bigg]_{1}^{2}=\frac{26}{3}\]

\[\int_{1}^{2}f(t)dt=\frac{t^{3}}{3}\bigg]_{1}^{2}=\frac{26}{3}\]

\[\int_{1}^{2}f(y)dy=\frac{y^{3}}{3}\bigg]_{1}^{2}=\frac{26}{3}\]

`Definite integral`² With Variable Upper Limit

If we have an integral¹ of the form

\[\int^x_a\]

Then we have to use different letter for the variable of integration¹ to differentiate it from the upper limit. Such as

\[\int_{2}^{x}t^{2}dt\]

The Theorem

\[A(x)=\int_{a}^{x}f(t)dt\]

\[\frac{d}{dx}(A(x))=A'(x)=\frac{d}{dx}\left(\int_{a}^{x}f(t)dt\right)=f(x)\]

If the integrand¹ is continuous,³ then the derivative⁴ of a definite integral² with respect to its upper limit is equal to the integrand¹ evaluated at the upper limit.

\[\frac{d}{dx}\left(\int_{a}^{x}f(t)dt\right)=f(x)\]

32. Second Fundamental Theorem of Calculus

Dummy Variable

Definite integral2 With Variable Upper Limit

The Theorem

References

`Definite integral`² With Variable Upper Limit