Evaluating integrals Evaluate the following integrals.

∫ 𝓍⁷ √(𝓍⁴ + 1d𝓍)

Guida verificata passo dopo passo

Step 1: Recognize that the integral involves a composite function. The term √(𝓍⁴ + 1) suggests a substitution method might simplify the integral. Let u = 𝓍⁴ + 1.

Step 2: Compute the derivative of u with respect to 𝓍. Since u = 𝓍⁴ + 1, we find du/d𝓍 = 4𝓍³. Rearrange to express du in terms of d𝓍: du = 4𝓍³ d𝓍.

Step 3: Rewrite the integral in terms of u. Substitute u = 𝓍⁴ + 1 and du = 4𝓍³ d𝓍 into the original integral. The integral becomes (1/4) ∫ 𝓍⁴ √u du.

Step 4: Simplify further. Notice that 𝓍⁴ can be expressed in terms of u using the substitution u = 𝓍⁴ + 1. Therefore, 𝓍⁴ = u - 1. Replace 𝓍⁴ in the integral to get (1/4) ∫ (u - 1) √u du.

Step 5: Break the integral into two simpler parts. Expand (u - 1) √u into u^(3/2) - u^(1/2) and integrate each term separately. Use the power rule for integration: ∫ u^n du = (u^(n+1))/(n+1) + C, where n ≠ -1.

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