Step-by-Step Examples

Each result below is checked by differentiation or substitution where appropriate. For an indefinite integral, the final answer represents a family of antiderivatives on an interval where the integrand is defined.

Example

Find the indefinite integral

∫x5 dx.\int x^5\,dx.

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Example

Evaluate

∫(3x2−4x+5) dx.\int(3x^2-4x+5)\,dx.

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Example

Determine

∫(2cos⁡x−3sin⁡x) dx.\int(2\cos x-3\sin x)\,dx.

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Example

Evaluate

∫(4ex+5x−2x) dx.\int\left(4e^x+\frac{5}{x}-2^x\right)\,dx.

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Example

Evaluate

∫(31−x2−21+x2) dx.\int\left(\frac{3}{\sqrt{1-x^2}}-\frac{2}{1+x^2}\right)\,dx.

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Substitution Examples

Example

Evaluate

∫2xcos⁡(x2) dx.\int 2x\cos(x^2)\,dx.

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Initial Value Problems and Applications

Example

A particle has acceleration a(t)=6t+4a(t)=6t+4, initial velocity v(0)=−6v(0)=-6 cm/s, and initial position s(0)=9s(0)=9 cm. Find s(t)s(t).

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Example

Suppose P′(t)=4t+3P'(t)=4t+3 and P(0)=50P(0)=50. Find P(t)P(t).

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Example

A company's marginal cost is C′(x)=3x2−20x+50C'(x)=3x^2-20x+50 dollars per item, and its fixed cost is 500 dollars, so C(0)=500C(0)=500. Find C(x)C(x) and C(10)C(10).

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Separable Differential Equation Example

Example

Solve

dydx=xy,y(0)=3.\frac{dy}{dx}=xy,\qquad y(0)=3.

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Key Takeaways
  • Differentiate every proposed antiderivative when practical; it is the fastest independent correctness check.
  • Domain restrictions matter for logarithmic and inverse-trigonometric antiderivatives.
  • In substitution, du=g′(x) dxdu=g'(x)\,dx encodes the Chain Rule rather than a claim that dxdx is a literal infinitesimal number.
  • Successive integrations introduce independent constants that are fixed by independent initial conditions.
  • Before dividing by a factor involving the dependent variable in a separable equation, check whether setting that factor to zero gives an equilibrium solution.