Basic Integrals

22 flashcards · Mathematics · by Elsa Schutfort

1. ∫sec2x\int_{}^{} sec^{2}x∫​sec2x

   tan(x)

2. ∫csc2x\int_{}^{} csc^{2}x∫​csc2x

   -cot(x)

3. ∫sec(x)tan(x)\int_{}^{} sec(x)tan(x)∫​sec(x)tan(x)

   sec(x)

4. ∫csc(x)cot(x)\int_{}^{} csc(x)cot(x)∫​csc(x)cot(x)

   -csc(x)

5. ∫tan(x)\int_{}^{} tan(x)∫​tan(x)

   ln|sec(x)|

6. ∫cot(x)\int_{}^{} cot(x)∫​cot(x)

   ln|sin(x)|

7. ∫sin2x\int sin^2 x∫sin2x

   1/2 u + 1/4sin2x

8. ∫cos2x\int cos^2 x∫cos2x

   12x+14sin2x\frac{1}{2} x + \frac{1}{4} sin2x 21​x+41​sin2x

9. ∫tan2x\int tan^2x∫tan2x

   tanx−xtan x - xtanx−x

10. ∫cot2x\int cot^2x∫cot2x

    −cotx−x-cotx - x−cotx−x

11. sinxsin xsinx

    cosxcosxcosx

12. Integration by Parts

    ∫udv=uv−∫vdu\int u dv = uv - \int v du∫udv=uv−∫vdu

13. sin(ax)cos(bx)

    12sin((a−b)x)+12sin((a+b)x)\frac{1}{2} sin((a-b)x) + \frac{1}{2} sin((a+b)x)21​sin((a−b)x)+21​sin((a+b)x)

14. Midpoint Rule

    Mn=∑i=1nf(mi)ΔxM_n = \sum_{i=1}^{n} f(mi) \Delta xMn​=∑i=1n​f(mi)Δx

15. Trapezoidal Rule

    Tn=12f(x0)+2f(x1)+2f(x2)+...+2f(xn−1)+f(xn))T_n = \frac{1}{2} f(x_0) + 2f(x_1)+2f(x_2) + ... + 2f(x_{n-1}) + f(x_n))Tn​=21​f(x0​)+2f(x1​)+2f(x2​)+...+2f(xn−1​)+f(xn​))

16. Midpoint Error

    Mn≤M(b−a)324n2M_{n} \leq \frac{M(b-a)^3}{24n^2}Mn​≤24n2M(b−a)3​

17. Trapezoidal Error

    Tn≤M(b−a)312n2T_n \leq \frac{M(b-a)^3}{12n^2}Tn​≤12n2M(b−a)3​

18. Simpson's Rule

    Sn=Δx3(f(x0)+4f(x1)+2f(x2)...+2f(xn−2)+4f(xn−1)+f(xn))S_n = \frac{\Delta x}{3} (f(x_0) + 4f(x_1) + 2f(x_2) ... + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n))Sn​=3Δx​(f(x0​)+4f(x1​)+2f(x2​)...+2f(xn−2​)+4f(xn−1​)+f(xn​))

19. Simpson's error

    Sn≤M(b−a)5180n4S_n \leq \frac{M(b-a)^5}{180n^4}Sn​≤180n4M(b−a)5​

20. ∫1xdx\int \frac{1}{x} dx∫x1​dx

    ln |x|

21. ∫sinx\int sinx∫sinx

    −cosx-cosx−cosx

22. ∫cosx\int cos x∫cosx

    sinxsin xsinx