Antiderivatives and Definite integrals

Antiderivatives

100

∫(5𝑥⁴ +8𝑥³ −9𝑥² −1)𝑑𝑥

𝑥⁵ + 2𝑥⁴ −3𝑥³ − 𝑥 + 𝐶

200

∫(𝑥^1/4 + 1/√𝑥 +1/𝑥 − 1/𝑥²)𝑑𝑥

(4/5)x^5/4 - 2√x + ln lxl + 1/𝑥 + c

300

∫(𝑒^𝑥 +sec𝑥tan𝑥−tan𝑥) 𝑑𝑥

𝑒^𝑥 +sec𝑥+ln|cos𝑥|+𝐶

400

 ∫(3^𝑥 + 4/(1+𝑥²))𝑑𝑥

(1/(ln3))3^𝑥 +4tan⁻¹ 𝑥+𝐶

500

∫ 5𝑥³/(1+𝑥⁴) 𝑑𝑥

(5/4)ln|1+𝑥⁴|+𝐶

U-Substitution

100

∫Cos(5x+3)(5) dx 

Sin(5x+3)+C

200

∫(4x⁶+7)¹²(24x⁵) dx

(4x⁶+7)¹³/13  +C

300

∫(7x+4)⁹ dx

(1/7)(7x+4)¹⁰/10 + c

400

∫1/(1+(4x)²) dx

1/4 tan^-1(4x)+C

500

∫sec⁷x tan x dx

(sec x)⁷/7 + C

Definite Integrals

100

∫_-6⁸ 10 dx

140

200

∫₀² (4x³-3x²+2x) dx

12

300

∫_-2^-1(6/x³) dx

-9/4

400

∫₀² (x/(1+2x²)² dx

2/9

500

∫₀^𝜋 (sec²(x/3)) dx

3√3

Particular solutions(Finding c)

100

dy/dx = x-4; y(2)=0

y = x²/2 - 4x + 6

200

𝑑𝑦/𝑑𝑥 = 3/𝑥²; 𝑦(1)=2

𝑦=−3/𝑥 +5

300

y'=x²-1; y(3)=7

y=(x³/3)-x+1

400

𝑦′ =sin𝑥; 𝑦(0)=5

𝑦=−cos𝑥+6

500

dy/dx=2/(x+1); y(0)=2

2ln|𝑥+1|+2

Bonus

100

∫_-14⁷ (7e^x) dx

7e⁷-7e^-14

200

∫ 2/(1−𝑥²)^1/2 𝑑𝑥

2sin⁻¹ 𝑥+𝐶

300

∫8 sin(5x) dx

(-8/5) cos(5x) + C

400

𝑑𝑧/𝑑𝑡 =cos𝑡−2sin𝑡; 𝑧(0)=5

𝑧(𝑡)=sin𝑡+2cos𝑡+3

500

d²y/ dx²= 4-3x ;y'(0)=5 & y(0)=7

y'=4x-((3x²)/2)+5

y''=2x²-(1/2)x³+5x+7