Why is only one integration constant written after integrating both sides?

Conditions

• Both sides of the equation are being integrated indefinitely.
• The equation has been successfully separated into P-terms and t-terms.

Reasoning, step by step

1. Integrate the left side with respect to P, which could theoretically add a constant.
2. Integrate the right side with respect to t, which also adds a constant.
3. Recognize that subtracting the left-side constant from the right-side constant yields a single net constant.
4. Conclude that keeping only one constant, such as $C_1$ on the right side, is mathematically equivalent and simpler.

Example

After integrating ($1/P$)dP = k dt, the speaker notes that although one could write constants on both sides, they can be combined, so it is enough to keep a single constant, labeled $C_1$, on the right, resulting in ln|P| = kt + $C_1$.

Common misconceptions

• Believing that two separate integration constants must be kept and tracked independently.
• Thinking that omitting a constant on one side loses information, when in fact the arbitrary constant absorbs the difference.

Watch the explanation

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