Math / Calculus

QuestionFind $\int\left(2 x^{2}+5 x^{3}\right) d x$ $+C$

Studdy Solution

STEP 1

1. The problem requires finding the indefinite integral of the polynomial function $2x^2 + 5x^3$ .
2. The integration will be performed term by term.
3. The constant of integration, $C$ , will be added at the end.

STEP 2

1. Integrate each term separately.
2. Add the constant of integration.

STEP 3

Integrate the first term $2x^2$ :
The power rule for integration states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ . Applying this rule:
$\int 2x^2 \, dx = 2 \cdot \frac{x^{2+1}}{2+1} = 2 \cdot \frac{x^3}{3} = \frac{2}{3}x^3$

STEP 4

Integrate the second term $5x^3$ :
Again, using the power rule for integration:
$\int 5x^3 \, dx = 5 \cdot \frac{x^{3+1}}{3+1} = 5 \cdot \frac{x^4}{4} = \frac{5}{4}x^4$

STEP 5

Combine the results of the integrations and add the constant of integration $C$ :
$\int (2x^2 + 5x^3) \, dx = \frac{2}{3}x^3 + \frac{5}{4}x^4 + C$
The indefinite integral of the function is:
$\boxed{\frac{2}{3}x^3 + \frac{5}{4}x^4 + C}$

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