technique to integrate

Question

Select the best technique to integrate each function.

1. ∫𝑥𝑒𝑥𝑑𝑥
2.  ∫𝑥336-𝑥2𝑑𝑥
3. ∫𝑥6(3𝑥+8)𝑑𝑥
4. ∫𝑥5(6-𝑥6)3𝑑𝑥
5.  ∫𝑥+8𝑥2+9𝑥+18𝑑𝑥
6. Algebraic manipulation or direct integration
7. Substitution
8. Integration by parts
9. Trig substitution
10. Partial fractions

 Please show work/reasoning if possible

Answer 1

Integral of xe^x dx

= xe^x - e^x + C

use integral tables, Integral of ue^u du = ue^u -e^u + C

check the answer

derivative of xe^x -e^x+ C (derivative of product yz = yz' + zy', derivative of e^x = e^x)

= xe^x + e^x - e^x

= xe^x

Integral of x^336 - x^2 dx

= (x^337)/337 - (x^3)/3 + C

power rule

increase power by 1 and divide by that number, for indefinite integral,always add C for unknown Constant

Integral of x^6(3x+8)dx

= Integral of 3x^7+ 8x^6 dx

= (3/8)x^8 + (8/7)x^7 + C

Integral of x^5(6-x^6)dx

= Integral of 6x^5 - x^11 dx

= (6/6)x^6 - (1/12)x^12 + C

= x^6 - (x^12)/12 + C

Integral of x + 8x^2 + 9x +18 dx

= Integral of 8x^2 +10x + 18 dx

= (8/3)x^3 +5x^2 + 18x + C