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Math  /  Calculus

QuestionВычислить неопределенные интегралы, сводя их к табличным:
1. ∫(2x5−3x2)dx=x63−x3+C\int\left(2 x^{5}-3 x^{2}\right) d x=\frac{x^{6}}{3}-x^{3}+C

Studdy Solution

STEP 1

1. We are given an indefinite integral to solve: ∫(2x5−3x2) dx\int (2x^5 - 3x^2) \, dx.
2. The goal is to simplify the integral into a form that can be recognized as a standard integral.
3. We will use basic rules of integration, particularly the power rule for integration.

STEP 2

1. Break down the integral into simpler parts.
2. Apply the power rule of integration to each term.
3. Combine the results and include the constant of integration.

STEP 3

Break down the integral into separate integrals for each term:
∫(2x5−3x2) dx=∫2x5 dx−∫3x2 dx\int (2x^5 - 3x^2) \, dx = \int 2x^5 \, dx - \int 3x^2 \, dx

STEP 4

Apply the power rule of integration to each term. The power rule states that ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.
For the first term:
∫2x5 dx=2⋅x5+15+1=2⋅x66=x63\int 2x^5 \, dx = 2 \cdot \frac{x^{5+1}}{5+1} = 2 \cdot \frac{x^6}{6} = \frac{x^6}{3}
For the second term:
∫3x2 dx=3⋅x2+12+1=3⋅x33=x3\int 3x^2 \, dx = 3 \cdot \frac{x^{2+1}}{2+1} = 3 \cdot \frac{x^3}{3} = x^3

STEP 5

Combine the results from each term and include the constant of integration C C :
∫(2x5−3x2) dx=x63−x3+C\int (2x^5 - 3x^2) \, dx = \frac{x^6}{3} - x^3 + C
The indefinite integral is:
x63−x3+C\boxed{\frac{x^6}{3} - x^3 + C}

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