Math Problem Statement
Реши задание
Find the integral: ∫(√x - 3)·(x² + √x - 2) dx.
Solution
Let's solve the integral in the image step by step. The problem asks us to compute:
Step 1: Expand the integrand
Multiply and :
= \sqrt{x} \cdot x^2 + \sqrt{x} \cdot \sqrt{x} - \sqrt{x} \cdot 2 - 3 \cdot x^2 - 3 \cdot \sqrt{x} + 3 \cdot 2.$$ Simplify each term: $$= x^{5/2} + x - 2\sqrt{x} - 3x^2 - 3\sqrt{x} + 6.$$ Combine like terms: $$= x^{5/2} + x - 5\sqrt{x} - 3x^2 + 6.$$ --- ### Step 2: Integrate term by term The integral becomes: $$\int \left( x^{5/2} + x - 5\sqrt{x} - 3x^2 + 6 \right) \, dx.$$ **1. Integrate $$x^{5/2}$$:** $$\int x^{5/2} \, dx = \frac{2}{7} x^{7/2}.$$ **2. Integrate $$x$$:** $$\int x \, dx = \frac{x^2}{2}.$$ **3. Integrate $$-5\sqrt{x}$$:** $$\int -5\sqrt{x} \, dx = \int -5x^{1/2} \, dx = -\frac{10}{3}x^{3/2}.$$ **4. Integrate $$-3x^2$$:** $$\int -3x^2 \, dx = -x^3.$$ **5. Integrate $$6$$:** $$\int 6 \, dx = 6x.$$ --- ### Step 3: Combine all terms Combine the results of each integral: $$\frac{2}{7} x^{7/2} + \frac{x^2}{2} - \frac{10}{3} x^{3/2} - x^3 + 6x + C.$$ --- ### Step 4: Compare with the options Looking at the choices in the image, the correct option matches: $$\boxed{\frac{2}{7}x^{7/2} - x^3 + \frac{x^2}{2} - \frac{10}{3}x^{3/2} + 6x + C.}$$ Let me know if you'd like further clarifications or more details! 😊 --- ### Related Questions: 1. What is the importance of expanding the integrand first before integration? 2. How do we handle fractional exponents in integration? 3. What are the common integration rules used in polynomial integrals? 4. How can substitution simplify certain integrals involving roots? 5. Why is the constant of integration $$C$$ important? **Tip:** Always simplify the integrand before integration to make the process more straightforward.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Integration
Polynomial Expansion
Integration of Power Functions
Formulas
∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1
Distributive Property: (a + b)(c + d) = ac + ad + bc + bd
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 10-12 or college level