Integrals in math

Question

Evaluate the following integrals:

A. ∫ (x^3 + 2x^2 + 1) dx

B. ∫ [2/(x + 5)] dx

C. ∫ (2e^2x) dx

Answer 1

All answers are bolded

Part A

∫(x3 + 2x2 +1)dx = (1/4)x⁴ + (2/3)x³ +x + C

this is the power rule, where ∫xⁿ = (xⁿ⁺¹)/(n+1) + C

Part B

Rewrite it as ∫2(x+5)^-1 dx = 2∫(x+5)^-1 dx

∫x^-1 dx = ln(x) + C

Answer is 2ln(x+5) + C

Part C

∫2e^2x dx = 2∫e^2x dx

To find ∫e^2x dx

Let g(x) = 2x

Then g'(x) = 2

Work in reverse with the chain rule

2e^2x = f'[g(x)]g'(x)=2[f'(2x)]

Divide both sides by 2

e^2x = f'(2x)

Therefore, f '(x) = e^x

Then f(x) = e^x

∫f'[g(x)]g'(x) = f[g(x)]

∫2e^2x dx = e^2x +C

Always check yourself by finding the derivative of your answers and comparing that with what is in the question.