Evaluating integrals Evaluate the following integrals.

∫₀¹ 𝓍 • 2ˣ²⁺¹ d𝓍

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Step 1: Recognize that the integral involves a product of functions, specifically 𝓍 and 2 raised to the power of (𝓍² + 1). This suggests that substitution might be a useful method to simplify the integral.

Step 2: Let u = 𝓍² + 1. Then, compute the derivative of u with respect to 𝓍: du/d𝓍 = 2𝓍. Rearrange this to express du in terms of d𝓍: du = 2𝓍 d𝓍.

Step 3: Substitute u and du into the integral. Replace 𝓍² + 1 with u and 𝓍 d𝓍 with (1/2) du. The integral becomes ∫₀¹ 𝓍 • 2ˣ²⁺¹ d𝓍 = (1/2) ∫ 2ᵘ du.

Step 4: Adjust the limits of integration to match the substitution. When 𝓍 = 0, u = 0² + 1 = 1. When 𝓍 = 1, u = 1² + 1 = 2. The integral now has limits from u = 1 to u = 2.

Step 5: Integrate 2ᵘ with respect to u. Recall that the integral of an exponential function aᵘ is (aᵘ / ln(a)), where a is the base of the exponential. Apply this formula to complete the integration, and evaluate the result at the new limits u = 1 and u = 2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as ∫_a^b f(x) dx, where 'a' and 'b' are the limits of integration. The result of a definite integral is a numerical value that quantifies the accumulation of the function's values between these two points.

Integration Techniques

Integration techniques are methods used to evaluate integrals that may not be solvable by basic antiderivatives. Common techniques include substitution, integration by parts, and recognizing patterns in integrals. For the integral ∫ 2x²+1 dx, applying the appropriate technique is crucial for finding the correct antiderivative.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base of the exponential, and 'x' is the exponent. In the context of the integral provided, understanding how to manipulate and integrate functions involving exponents, such as 2^(x²+1), is essential for evaluating the integral correctly.

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