The length of the curve y = x⁴ from x = 1 to x = 5 is given by?

We will be using the arc length formula to get the exact length of the curve and solve this.

Answer: The length of the curve y = x⁴ from x = 1 to x = 5 is ln|500 + 53√89| - ln|4 + √17|.

Let's solve this step by step.

Explanation:

Given, y = x⁴ , x = 1 to x = 5

Length of the curve y = f(x) from x = a to x = b is given by: \(\int_{a}^{b}\) √1 + [f′(x)]² dx

Let's find the first derivative of y = x⁴.

y = x⁴ 

dy/dx = 4x³ 

Length of curve =\(\int_{1}^{5}\) √[1 + {f′(x)}]² dx

=\(\int_{1}^{5}\) √[1 + (4x³)²] dx

= \(\int_{1}^{5}\) √[1 + 16x⁶] dx

Substitue 4x³ = tanθ in the indefinite integral  ∫ √[1 + 16x⁶] dx:

= ∫ √[1 + tan²θ] dx

= ∫ √sec²θ dx

= ∫ secθ dx

= ln|tanθ + secθ|

Resubstitute the values: tanθ = 4x³, and secθ = √[1 + 16x⁶]

= [ln|4x³ + √1 + 16x⁶ |]\(^5_1\)

= [ln|4(5)³ + √{1 + 16(5)⁶}|] - [ln|4(1)³ + √{1 + 16(1)⁶}|]

= [ln|4(125) + √{1 + 16(15625)}|] - [ln|4 + √{1 + 16}|]

= [ln|500 + √250001|] - [ln|4 + √17|]

= ln|500 + 53√89| - ln|4 + √17|

Hence, the length of the curve y = x⁴ from x = 1 to x = 5 is ln|500 + 53√89| - ln|4 + √17|.