Step-by-Step Examples

Here are several examples demonstrating how to carefully apply linearity properties and basic rules to evaluate indefinite integrals. Always remember to append +C+ C to your final expression!

Example

Find the indefinite integral:
∫x5 dx \int x^5 \, dx

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Example

Evaluate the following polynomial integral:
∫(3x2−4x+5) dx \int (3x^2 - 4x + 5) \, dx

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Example

Determine the indefinite integral involving trigonometric functions:
∫(2cos⁡x−3sin⁡x) dx \int (2\cos x - 3\sin x) \, dx

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Example

Evaluate the indefinite integral involving exponential and logarithmic forms:
∫(4ex+5x−2x) dx \int \left( 4e^x + \frac{5}{x} - 2^x \right) \, dx

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Example

Evaluate the indefinite integral involving inverse trigonometric forms:
∫(31−x2−21+x2) dx \int \left( \frac{3}{\sqrt{1 - x^2}} - \frac{2}{1 + x^2} \right) \, dx

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Key Takeaways
  • Always remember to include the constant of integration, +C+ C, in your final answer for indefinite integrals.
  • Combine multiple constants of integration into a single constant at the end of the evaluation.
  • Ensure you are comfortable with basic exponential, logarithmic, and inverse trigonometric integration formulas as they frequently appear alongside polynomials.

Initial Value Problems Examples

Example

A particle moves along a straight line with an acceleration given by a(t)=6t+4a(t) = 6t + 4. Its initial velocity at t=0t = 0 is v(0)=−6v(0) = -6 cm/s, and its initial position is s(0)=9s(0) = 9 cm. Find its position function s(t)s(t).

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Example

Suppose the rate of change of a population is given by P′(t)=4t+3P'(t) = 4t + 3. If the initial population at time t=0t=0 is 5050, find the population function P(t)P(t).

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Example

A company's marginal cost for producing xx items is C′(x)=3x2−20x+50C'(x) = 3x^2 - 20x + 50 (in dollars per unit). If the fixed costs (cost when x=0x=0) are \500,findthetotalcostfunction, find the total cost function C(x)$ and determine the cost of producing 10 items.

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