Fundamental theorem of calculus

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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)
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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