AS Pure Chapter 13: Integration

6 flashcards · Mathematics · by TheoTyson

1. If f′(x)=xnf'(x) = x^nf′(x)=xn , then what does f(x)f(x)f(x) equal?

   f(x)=1n+1xn+1+cf(x) = \frac{1}{n+1} x^{n+1} + cf(x)=n+11​xn+1+c

2. If f′(x)=kxnf'(x) = kx^nf′(x)=kxn , then what does f(x)f(x)f(x) equal?

   f(x)=kn+1xn+1+cf(x) = \frac{k}{n+1} x^{n+1} + cf(x)=n+1k​xn+1+c

3. What is ∫f′(x) dx\int{f'(x)} \, \text{d} x∫f′(x)dx equal to?

   ∫f′(x) dx=f(x)+c\int{f'(x)} \, \text{d} x = f(x) + c∫f′(x)dx=f(x)+c

4. Expand ∫(f(x)+g(x)) dx\int{(f(x) + g(x))} \, \text{d} x∫(f(x)+g(x))dx

   ∫(f(x)+g(x)) dx=∫f(x) dx+∫g(x) dx\int{(f(x) + g(x))} \, \text{d} x = \int{f(x)} \, \text{d} x + \int{g(x)} \, \text{d} x∫(f(x)+g(x))dx=∫f(x)dx+∫g(x)dx

5. How do you solve ∫abf′(x) dx\int_{a}^{b} f'(x) \, \text{d} x∫ab​f′(x)dx ?

   ∫abf′(x) dx=[f(x)]ab=f(b)−f(a)\int_{a}^{b} f'(x) \, \text{d} x = [f(x)]_a^b = f(b) - f(a)∫ab​f′(x)dx=[f(x)]ab​=f(b)−f(a)

6. What is the area between the curve y=f(x)y = f(x)y=f(x) , the xxx -axis, the lines x=ax=ax=a and x=bx=bx=b ?

   Area=∫aby dxArea = \int_{a}^{b} y \, \text{d} xArea=∫ab​ydx