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# The length of the curve y = x^{4} 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 = x^{4} from x = 1 to x = 5 is ln|500 + 53√89| - ln|4 + √17|.

Let's solve this step by step.

**Explanation:**

Given, y = x^{4} , 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 = x^{4}.

y = x^{4}

dy/dx = 4x^{3}

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

=\(\int_{1}^{5}\) √[1 + (4x^{3})^{2}] dx

= \(\int_{1}^{5}\) √[1 + 16x^{6}] dx

Substitue 4x^{3} = tanθ in the indefinite integral ∫ √[1 + 16x^{6}] dx:

= ∫ √[1 + tan^{2}θ] dx

= ∫ √sec^{2}θ dx

= ∫ secθ dx

= ln|tanθ + secθ|

Resubstitute the values: tanθ = 4x^{3}, and secθ = √[1 + 16x^{6}]

= [ln|4x^{3} + √1 + 16x^{6} |]\(^5_1\)

= [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 = x^{4} from x = 1 to x = 5 is ln|500 + 53√89| - ln|4 + √17|.

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