intergrals calculus

Question

Find the formula for the function represented by the integral.

(Express numbers in exact form. Use symbolic notation and fractions where needed.)

∫𝑥−2(16𝑡^2−15𝑡)𝑑𝑡

Answer 1

Hi Maria,

Since the integral has "dt" at the end, we are integrating with respect to the variable "t" and all other variables, such as "x" we consider as constants. We start by splitting the integral into two parts,

∫x-2(16t²-15t)dt = ∫x dt + ∫-2(16t²-15t)dt. The first integral, we can do directly since "x" is treated as a constant: ∫x dt = x*t + c₁, where c₁ is an integration constant.

For the second integral, we expand out: ∫-2(16t²-15t)dt = ∫-32t² +30t dt then, we apply the rule for integrating poewrs (rule: when integrating a power of t, we add one to the exponent and divide by the new exponent).

∫-32t²-30t dt = -32 t⁽²⁺¹⁾/(2+1) +30 t⁽¹⁺¹⁾/(1+1) + c₂ = -32t³/3 +30t²/2 + c₂ = -32t³/3 +15t² + c₂, where c₂ is an integration constant.

Thus, the overall result is x*t + c₁ -32t³/3 +15t² + c₂. We can combine the two integration constants into one constant, giving the final result x*t -32t³/3 +15t² + c.

Hope this help!

Best,

Ibrahim