Evaluate the integrals in Exercises 31–78.
49. ∫x3^(x²)dx

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Identify the integral to be solved: $\int x 3^{x^{2}} \, dx$.

Recognize that the integrand contains a composite function \$3^{x^{2}}$ multiplied by $x$, suggesting a substitution related to the exponent $x^{2}$.

Let $u = x^{2}$. Then, compute the differential $du = 2x \, dx$, which implies $x \, dx = \frac{du}{2}$.

Rewrite the integral in terms of $u$: $\int x 3^{x^{2}} \, dx = \int 3^{u} \cdot \frac{du}{2} = \frac{1}{2} \int 3^{u} \, du$.

Recall the formula for integrating exponential functions with base $a$: $\int a^{u} \, du = \frac{a^{u}}{\ln(a)} + C$. Use this to express the integral in terms of $u$ and then substitute back $u = x^{2}$.

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Key Concepts

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

Integration with Variable Limits

When the integrand involves expressions like x raised to the power of x², it often requires recognizing the function's form and applying appropriate integration techniques, as the exponent itself is a function of x. Understanding how to handle variable exponents is crucial.

Techniques of Integration (Substitution and Logarithmic Differentiation)

For integrals involving functions like x^(x²), rewriting the expression using exponentials and logarithms (e.g., x^(x²) = e^(x² ln x)) allows the use of substitution and differentiation under the integral sign. This approach simplifies the integral into a more manageable form.

Integration by Parts

Integration by parts is a method used when the integrand is a product of functions, such as x multiplied by another function. It transforms the integral into simpler parts, often reducing the problem to a solvable integral or a recursive relation.

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