# Fundamental theorem of calculus

### Fundamental theorem of calculus

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.

#### Lessons

• 1.
Overview:
a)
If $f$ is continuous on $\left[ {a,b} \right]$, then:
$\frac{d}{{dx}}\int_a^x f\left( t \right)dt = f\left( x \right)$

b)
If $f$ is continuous on $\left[ {a,b} \right]$, then:
$\int_a^b f\left( x \right)dx = F\left( b \right) - F\left( a \right)$

• 2.
Fundamental Theorem of Calculus Part I
Evaluate.
a)
$\frac{d}{{dx}}\int_{1000}^x \sqrt {5 + 8t\;} dt$

b)
$\frac{d}{{dx}}\int_{ - 10}^{{x^6}} \frac{{{{\sin }^2}\left( {5{t^3} - t + 8} \right)}}{{{e^{4t}}}}dt$

• 3.
Fundamental Theorem of Calculus Part II
Evaluate.
a)
$\int_{ - 1}^3 {x^2}dx$

b)
$\int_{ - 1}^4 \frac{5}{{{x^2}}}dx$

c)
$\int_1^{{e^2}} \frac{5}{x}dx$

d)
$\int_{\frac{\pi }{5}}^\pi 4\cos \left( {5\theta } \right)d\theta$

e)
$\int_{ln5}^{ln7} 9{e^x}dx$