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Evaluate the indefinite integral \int x(4+3x^4)dx

Question:

Evaluate the indefinite integral {eq}\int x(4+3x^4)dx {/eq}

Indefinite Integrals:

When there are no bounds present on the indefinite integral, the anti-derivative of the expression cannot be evaluated at inputs. Therefore, a constant of integration is added to the anti-derivative to show the general solution of the expression under the indefinite integral.

Answer and Explanation:

Given: {eq}\int x(4+3x^4) dx {/eq}


To evaluate the indefinite integral, the strategy is to take the anti-derivative of the expression before adding a constant of integration. To do this, the opposite operations of the power rule for derivatives will be applied, which means the current exponent for a specific term will be added by {eq}1 {/eq} before dividing the term by the new exponent value.


{eq}\begin{align*} \int x(4+3x^4) dx &= \int 4x+3x^5 dx \text{ [Distribution of the expression within the integrand]} \\ &= \frac{4x^{1+1}}{1+1}+\frac{3x^{5+1}}{5+1}+C \text{ [Constant of integration added to anti-derivative]} \\ &= \frac{4x^2}{2}+\frac{3x^6}{6}+C \\ &= 2x^2+\frac{x^6}{2}+C \\ \end{align*} {/eq}


Learn more about this topic:

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Indefinite Integrals as Anti Derivatives

from Math 104: Calculus

Chapter 12 / Lesson 11
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