Fundamental theorem of calculus

All in One Place

Everything you need for JC, LC, and college level maths and science classes.

Learn with Ease

We’ve mastered the national curriculum so that you can revise with confidence.

Instant Help

24/7 access to the best tips, walkthroughs, and practice exercises available.

0/2
?
Intros
Lessons
  1. Overview:
  2. If ff is continuous on [a,b]\left[ {a,b} \right], then:
    ddxaxf(t)dt=f(x)\frac{d}{{dx}}\int_a^x f\left( t \right)dt = f\left( x \right)
  3. If ff is continuous on [a,b]\left[ {a,b} \right], then:
    abf(x)dx=F(b)F(a)\int_a^b f\left( x \right)dx = F\left( b \right) - F\left( a \right)
0/7
?
Examples
Lessons
  1. Fundamental Theorem of Calculus Part I
    Evaluate.
    1. ddx1000x5+8t  dt\frac{d}{{dx}}\int_{1000}^x \sqrt {5 + 8t\;} dt
    2. ddx10x6sin2(5t3t+8)e4tdt\frac{d}{{dx}}\int_{ - 10}^{{x^6}} \frac{{{{\sin }^2}\left( {5{t^3} - t + 8} \right)}}{{{e^{4t}}}}dt
  2. Fundamental Theorem of Calculus Part II
    Evaluate.
    1. 13x2dx\int_{ - 1}^3 {x^2}dx
    2. 145x2dx\int_{ - 1}^4 \frac{5}{{{x^2}}}dx
    3. 1e25xdx\int_1^{{e^2}} \frac{5}{x}dx
    4. π5π4cos(5θ)dθ\int_{\frac{\pi }{5}}^\pi 4\cos \left( {5\theta } \right)d\theta
    5. ln5ln79exdx\int_{ln5}^{ln7} 9{e^x}dx
Topic Notes
?
In this lesson, we will learn about part 1 and part 2 of the Fundamental Theorem of Calculus. In part 1, we see that taking the derivative of an integral will just result in giving us the original function. However in some cases, we get the original function AND the derivative of the upper limit. Lastly in part 2, we will learn another way of evaluating the definite integral. To evaluate the definite integral, we must take the difference of the anti-derivative of the function at the upper limit, and the anti-derivative of the function at the lower limit. We will apply this theorem to many types of definite integrals such as polynomial integrals, trigonometric integrals, logarithmic integrals, and exponential integrals.