Calculus

U-Substitution

100

∫-2(√9-x²) dx

(2/3)(9-x²)^3/2+c

200

∫x(4x²+3)²dx

(1/32)(4x^2+3)^4+c

300

∫(x²/(1+x³)2)dx

(-1/3)(1+x^3)^2+c

400

∫tan⁴Øsec²ØdØ

tan^4Øsec^2Ø

500

∫(sinØ/cos²Ø)dØ

cos^2Ø

Indefinite Integrals

100

∫(4x+3)dx

2x^2+3x+C

200

∫(4x²-8x+1)dx

(4x^3/3)-4x^2+x+C

300

∫(9x²-4x+3)dx

3t^3-2t^2+x+C

400

∫(-1/5)sin u du

(1/5)cosu+C

500

∫(cscx cotx secx)dx

-cotx+C

Definite Integrals

100

∫₀¹2xdx

1

200

∫₂⁷3dx

15

300

∫_-1⁰(x-2)dx

-5/2

400

∫₂⁵(-3x+4)dx

-39/2

500

∫₀^pi(1+sinx)dx

Pi+2

Implicit Differentiation dy/dx

100

3x^2+5y^2=-y+2

-6x/10y+1

200

y=4x^2+5y^3

8x/1-15y^2

300

-2y+2=2x^3+3x^2

3x^2/1+3y

400

4y^3=2x^3+3xy^3

2x^2+y^3/4y^2-3y^2x

500

cos3y^2=4x+2

-2/sin(3y^2)3y

L’Hospital’s Rule

100

Lim_x->2(x^3-8/x^2-4)

3

200

Lim_x->0(sin5x/x)

5

300

Lim_x->2(x^2-4x+4/x^3-12x+16)

1/4

400

Lim_x->pi/2(1-sinx/1+cos2x)

1/4

500

Lim_x->3(x-4/x-2)

-1