Derivative of an Integral

The derivative of an integral is the result obtained by differentiating the result of an integral. Integration is the process of finding the "anti" derivative and hence by differentiating an integral should result in the original function itself. But this may not be the scenario with all definite integrals.

Let us learn more about the derivative of an integral (in different cases) along with more examples.

The derivative of an integral of a function is the function itself. But this is always true only in the case of indefinite integrals. The derivative of a definite integral of a function is the function itself only when the lower limit of the integral is a constant and the upper limit is the variable with respect to which we are differentiating. To summarize:

• The derivative of an indefinite integral of a function is the function itself. i.e., d/dx ∫ f(x) dx = f(x)
• The derivative of a definite integral with constant limits is 0. i.e., d/dx ∫_a ^b f(t) dt = 0
• The derivative of a definite integral where the lower limit is a constant and the upper limit is a variable is a function itself in terms of the given variable (upper bound).
  i.e., d/dx ∫_a^x f(t) dt = f(x) where 'a' is a constant and 'x' is a variable.

Let us look into each of these cases in detail. Also, let us see how to evaluate definite integrals that don't match with any of the last two cases.

Let us consider an indefinite integral ∫ x^3/2 dx. If we evaluate this using the power rule of integration, we get (2x^5/2)/5 + C. If we differentiate this using the power rule of differentiation, we get (2/5) (5/2) (x^3/2) + 0 = x^3/2. If we put this integration and differentiation in one step, we can write it as

d/dx ∫ x^3/2 dx = x^3/2.

i.e., the derivative of an indefinite integral is equal to the original function itself (it is a kind of "derivative and integral symbol get canceled with each other"). Thus, for any function f(x), we can write

d/dx ∫ f(x) dx = f(x)

A definite integral is of the form ∫_a ^b f(t) dt. But here, the limits always don't need to be constants. There can be 3 cases.

• Both limits can be constants.
• The lower limit is a constant and the upper limit is a variable.
• Both limits may involve variables.

We will study how to find the derivative of an integral in each of these cases.

When Both Limits are Constants

Consider the definite integral ∫_a ^b f(x) dx where both 'a' and 'b' are constants. Then by the second fundamental theorem of calculus, ∫_a ^b f(x) dx = F(b) - F(a) where F(x) = ∫ f(t) dt. Now, let us compute its derivative. d/dx∫_a ^b f(x) dx = d/dx [F(b) - F(a)] = 0 (as F(b) and F(a) are constants). Thus, when both limits are constants, the derivative of a definite integral is 0.

i.e., d/dx ∫_a ^b f(t) dt = 0

When One of the Limits is a Constant

Consider a definite integral ∫_a^x f(t) dt, where 'a' is a constant and 'x' is a variable. Then by the first fundamental theorem of calculus, d/dx ∫_a^x f(t) dt = f(x). This would reflect the fact that the derivative of an integral is the original function itself. Here are some examples.

• d/dx ∫₂^x t³ dt = x³.
• d/dx ∫_-1^x sin t² dt = sin x².

Note that here the lower limit should be a constant. If the upper limit is a constant and the lower limit is a variable, say d/dx ∫_x² t³ dt then we rewrite (using the property of definite integral) as - d/dx ∫₂^x t³ dt and now its result using the above examples is -x³.

When Both Limits Have Variables

Consider the integral ∫_t²^t³ log (x³ + 1) dx. Here, both the limits involve the variable t. In such cases, we apply a property of definite integral that says ∫_a^c f(t) dt = ∫_a^b f(t) dt + ∫_b^c f(t) dt and we assume 'b' to be a random constant while applying this property. Then we can write the above integral as

∫_t²^t³ log (x³ + 1) dx = ∫_t²¹ log (x³ + 1) dx + ∫₁^t³ log (x³ + 1) dx

Now, we apply another property of definite integrals that says ∫_a^b f(t) dt = - ∫_b^a f(t) dt. Using this, we can write ∫_t²¹ log (x³ + 1) dx as - ∫₁^t² log (x³ + 1) dx. Now the above step becomes:

= - ∫₁^t² log (x³ + 1) dx + ∫₁^t³ log (x³ + 1) dx

Taking the derivative on both sides,

d/dt ∫_t²^t³ log (x³ + 1) dx = - d/dt ∫₁^t² log (x³ + 1) dx + d/dt ∫₁^t³ log (x³ + 1) dx

For the first integral, assume that t² = u and for the second integral, assume that t³ = v. By the chain rule, we can write

= - [ d/du ∫₁^u log (x³ + 1) dx ] [ du/dt ] + [ d/dv ∫₁^v log (x³ + 1) dx ] [ dv/dt ]

= - [ d/du ∫₁^u log (x³ + 1) dx ] [ 2t ] + [ d/dv ∫₁^v log (x³ + 1) dx ] [ 3t² ]

Since the each integral has its lower limit to be a constant and upper limit to be a variable, their derivatives are equal to the functions in terms of the respective variables. i.e.,

= [- log (u³ + 1) ] [ 2t ] + [log (v³ + 1) ] [ 3t² ]

Substitute u = t² and v = t³ back here,

= - 2t log (t⁶ + 1) + 3t² log (t⁹ + 1)

This procedure is very helpful to find out the derivative of an integral.

Derivative of an Integral Formula

We have seen that the derivative of the integral ∫_t²^t³ log (x³ + 1) dx is - 2t log (t⁶ + 1) + 3t² log (t⁹ + 1) and this can be written as 3t² log (t⁹ + 1) - 2t log (t⁶ + 1). Note that the derivative of the upper limit t³ is 3t² and the derivative of the lower limit t² is 2t here. Thus, we can compute the derivative of an integral formula as follows:

∫_g(t)^h(t) f(x) dx = h'(t) · f(h(t)) - g'(t) · f(g(t))

where, f(h(t)) and f(g(t)) are the composite functions. i.e., to find the derivative of an integral:

• Step 1: Find the derivative of the upper limit and then substitute the upper limit into the integrand. Multiply both results.
• Step 2: Find the derivative of the lower limit and then substitute the lower limit into the integrand. Multiply both results.
• Step 3: Subtract the above results in order.

☛Related Topics:

• Definite Integral Calculator
• Integral Calculator
• Indefinite Integral Calculator