Find f'(x) if f(x) = integral_0^4x dt / t^5 + 1.


Find {eq}f'(x) {/eq} if {eq}\displaystyle f(x) = \int_0^{4x} \frac {dt}{t^5 + 1} {/eq}.

Fundamental Theorems of Calculus

There are two fundamental theorems of calculus. The first fundamental theorem states that:

{eq}\int_{a}^{b} f(x) dx = F(b)-F(a) {/eq}

where F(x) is the antiderivative of f(x). The second fundamental theorem states that given:

{eq}F(x)=\int_{a}^{x} f(t) dt {/eq}

we have the derivative given by {eq}F'(x)=f(x) {/eq}. As with the rules for differentiation, remembering these two theorems involving integrals can greatly simplify integration questions.

Answer and Explanation:

Become a member to unlock this answer! Create your account

View this answer

See full answer below.

Learn more about this topic:

The Fundamental Theorem of Calculus


Chapter 12 / Lesson 10

The fundamental theorem of calculus is one of the most important points to understand in mathematics. Learn to define the formula of the fundamental theorem of calculus and explore examples of it put into practice.

Related to this Question

Explore our homework questions and answers library