Calc formula quiz 3/5

Integral Formulas

1. ∫eu du=eu+C\int e^u \, du = e^u + C

2. ∫1u du=ln⁡∣u∣+C\int \frac{1}{u} \, du = \ln |u| + C

3. ∫sec⁡utan⁡u du=sec⁡u+C\int \sec u \tan u \, du = \sec u + C

4. ∫tan⁡u du=ln⁡∣sec⁡u∣+C\int \tan u \, du = \ln |\sec u| + C

5. ∫sec⁡u du=ln⁡∣sec⁡u+tan⁡u∣+C\int \sec u \, du = \ln |\sec u + \tan u| + C

6. ∫cos⁡u du=sin⁡u+C\int \cos u \, du = \sin u + C

7. ∫csc⁡2u du=−cot⁡u+C\int \csc^2 u \, du = -\cot u + C

8. ∫sec⁡2u du=tan⁡u+C\int \sec^2 u \, du = \tan u + C

9. ∫csc⁡ucot⁡u du=−csc⁡u+C\int \csc u \cot u \, du = -\csc u + C

10. ∫cot⁡u du=ln⁡∣sin⁡u∣+C\int \cot u \, du = \ln |\sin u| + C

11. ∫csc⁡u du=ln⁡∣csc⁡u−cot⁡u∣+C\int \csc u \, du = \ln |\csc u - \cot u| + C

12. ∫sin⁡u du=−cos⁡u+C\int \sin u \, du = -\cos u + C

13. ∫au du=auln⁡a+C\int a^u \, du = \frac{a^u}{\ln a} + C, where a>0a > 0, a≠1a \neq 1

Derivative Formulas

1. ddxln⁡u=1u⋅dudx\frac{d}{dx} \ln u = \frac{1}{u} \cdot \frac{du}{dx}

2. ddxlog⁡au=1uln⁡a⋅dudx\frac{d}{dx} \log_a u = \frac{1}{u \ln a} \cdot \frac{du}{dx}

3. ddxau=auln⁡a⋅dudx\frac{d}{dx} a^u = a^u \ln a \cdot \frac{du}{dx}

Fundamental Theorem of Calculus (FTC)

1. First FTC:
   ddx∫axf(t) dt=f(x)\frac{d}{dx} \int_{a}^{x} f(t) \, dt = f(x)

2. Second FTC:
   ∫abf(x) dx=F(b)−F(a)\int_{a}^{b} f(x) \, dx = F(b) - F(a)

3. Second FTC (Chain Rule Version):
   ddx∫g(x)h(x)f(t) dt=f(h(x))h′(x)−f(g(x))g′(x)\frac{d}{dx} \int_{g(x)}^{h(x)} f(t) \, dt = f(h(x)) h'(x) - f(g(x)) g'(x)

Average Value Theorem
f(c)=1b−a∫abf(x) dxf(c) = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx