10 mistakes learners make with integration, each one named and explained.
Raise the power by one, then divide by the new power.
The constant 3 stays as 3 instead of becoming 3x.
Constants vanish when differentiating, so it is easy to treat them as inert.
Question. integrate 3
A common answer. 3
The answer. 3x + c
Why. Integrating is the reverse, so a constant becomes a term in x.
Every term gains a power when integrated, including the ones that had none.
6x^2 integrates to 3x^3, dividing by 2 instead of 3.
The power in the question is the one in front of you when you divide.
Question. 6x^2
A common answer. 3x^3 + c
The answer. 2x^3 + c
Why. You divide by the power you have just created, not the one you started with.
Raise the power first and write it down. Then divide by the number you just wrote.
sin(3x) integrates to -3cos(3x), when it should be divided by 3.
The chain rule multiplies when differentiating, and integrating feels like the same operation.
Question. integrate sin(3x)
A common answer. -3cos(3x) + c
The answer. -(1/3)cos(3x) + c
Why. Differentiating -(1/3)cos(3x) gives sin(3x), which is the check.
Everything reverses when you integrate, including the direction of that step.
The integral is evaluated at the top limit and the bottom one is never taken off.
The upper limit is substituted first, and its value looks like an answer.
Question. integral of 3x^2 + 2 from 1 to 3
A common answer. 27 + 6 = 33
The answer. 33 - 3 = 30
Why. A definite integral is the value at the top minus the value at the bottom.
Write the square brackets with both limits before substituting. The subtraction is then already on the page.
sin integrates to sin, with only the coefficient adjusted.
The coefficient work is where the attention goes, and the function itself gets copied across.
Question. integrate sin(3x)
A common answer. -(1/3)sin(3x) + c
The answer. -(1/3)cos(3x) + c
Why. Integrating sin gives a cosine, because differentiating cosine is what produces a sine.
Sin and cos always swap. If yours did not, only the coefficient was handled.
1/(2x - 5) integrates to ln|2x - 5| with no factor of a half.
The logarithm is the part that has to be recalled, and it feels like the whole answer.
Question. integrate 1/(2x - 5)
A common answer. ln|2x - 5| + c
The answer. (1/2)ln|2x - 5| + c
Why. Differentiating ln|2x - 5| gives 2/(2x - 5), which is twice too big.
Differentiate your answer. Any spare factor tells you what you missed.
The lower limit's value is subtracted from nothing, or the two are reversed, giving a negative.
Two numbers and a subtraction, and the order is decided by the notation rather than by meaning.
Question. integral from 1 to 3
A common answer. value at 1 minus value at 3, giving -30
The answer. 30
Why. The upper limit always comes first in the subtraction.
An area under a positive curve cannot be negative. That is the check.
6x^2 integrates to 6x^3, with no division by the new power.
Raising the power is the visible reversal of differentiating, and the division is the quiet half.
Question. 6x^2
A common answer. 6x^3 + c
The answer. 2x^3 + c
Why. The new power is 3, so the coefficient is divided by 3.
Differentiate your answer back. It has to return exactly what you started with.
1/(2x - 5) is integrated by raising the power, giving a squared bracket.
The power rule is the default and 1 over something is a power of minus one.
Question. integrate 1/(2x - 5)
A common answer. (1/2)(2x - 5)^2 + c
The answer. (1/2)ln|2x - 5| + c
Why. Raising a power of -1 by one gives a power of 0, which is why the rule breaks down here.
Whenever the power is exactly -1, stop and reach for ln instead.
sin integrates to positive cos rather than negative cos.
Four sign rules across sin and cos, differentiating and integrating, and only one of them is wrong.
Question. integrate sin(3x)
A common answer. (1/3)cos(3x) + c
The answer. -(1/3)cos(3x) + c
Why. Differentiating cos gives minus sin, so integrating sin has to give minus cos.
Only one of the four carries a minus in each direction. Differentiate your answer to find out which.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.