Title: InkMaths / Integration: standard results
On every note of this deck, unless a note says otherwise:
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Front: CHECK YOURSELF
Find the integral of 6e^3x with respect to x.
Back: 2e^3x + c. Divide by the 3 the chain rule would have produced. An indefinite integral written without the constant loses the last mark, every time.

Front: STANDARD RESULT — LEARN IT
Sine (x in radians):
∫ sin x dx = ?\(\text{∫} \sin x \, dx = ?\)∫ sin x dx = ?
Back: ∫ sin x dx = −cos x + c\(\text{∫} \sin x \, dx = −\cos x + c\)∫ sin x dx = − cos x + c

Front: STANDARD RESULT — LEARN IT
Exponential (k ≠ 0):
∫ e^kx dx = ?\(\text{∫} e^{kx} \, dx = ?\)∫ e^kx dx = ?
Back: ∫ e^kx dx = 1ke^kx + c\(\text{∫} e^{kx} \, dx = \frac{1}{k}e^{kx} + c\)∫ e^kx dx = (1/k)e^kx + c

Front: STANDARD RESULT — LEARN IT
Log integral:
∫ f'(x)f(x) dx = ?\(\text{∫} \frac{\text{f}'(x)}{\text{f}(x)} \, dx = ?\)∫ (f′(x)/f(x)) dx = ?
Back: ∫ f'(x)f(x) dx = ln|f(x)| + c\(\text{∫} \frac{\text{f}'(x)}{\text{f}(x)} \, dx = \text{ln}|\text{f}(x)| + c\)∫ (f′(x)/f(x)) dx = ln| f(x)| + c

Front: STANDARD RESULT — LEARN IT
The n = −1 case:
∫ 1x dx = ?\(\text{∫} \frac{1}{x} \, dx = ?\)∫ (1/x) dx = ?
Back: ∫ 1x dx = ln|x| + c\(\text{∫} \frac{1}{x} \, dx = \text{ln}|x| + c\)∫ (1/x) dx = ln|x| + c

Front: RECALL
State what to do when part of an area lies below the x-axis.
Back: Split the integral at the root and take the modulus of each piece before adding. Integrating straight through returns the signed total, so the two parts cancel and the area comes out too small. Sketch first and the roots tell you where to split.

Front: STANDARD RESULT — LEARN IT
Area under a parametric curve:
∫ y dx = ? in terms of t\(\text{∫} y \, dx = ? \text{ in terms of t}\)∫ y dx = ? in terms of t
Back: ∫ y dxdt dt, with the limits converted to values of t\(\text{∫} y \frac{dx}{dt} \, dt, \text{ with the limits converted to values of t}\)∫ y (dx/dt) dt, with the limits converted to values of t

Front: RECALL
State how to solve a separable differential equation.
Back: Get every y with the dy and every x with the dx, then integrate both sides and write one constant. Dividing by a term in y assumes it is not zero, and any y that makes it zero is a constant solution in its own right. Find that constant from the given condition before you rearrange, because rearranging first turns c into something you cannot evaluate.

Front: STANDARD RESULT — LEARN IT
Cosine (x in radians):
∫ cos x dx = ?\(\text{∫} \cos x \, dx = ?\)∫ cos x dx = ?
Back: ∫ cos x dx = sin x + c\(\text{∫} \cos x \, dx = \sin x + c\)∫ cos x dx = sin x + c

Front: SPOT THE ERROR
Say what is wrong with this claim:
The integral of sin x is cos x.
Back: It is −cos x + c. Check by differentiating your answer: the derivative of −cos x is sin x, which recovers the integrand. Differentiate-to-check catches every sign slip of this kind.

Front: STANDARD RESULT — LEARN IT
Power rule (n ≠ −1):
∫ x^n dx = ?\(\text{∫} x^{n} \, dx = ?\)∫ x^n dx = ?
Back: ∫ x^n dx = x^n+1n+1 + c\(\text{∫} x^{n} \, dx = \frac{x^{n+1}}{n+1} + c\)∫ x^n dx = x^(n + 1)/(n + 1) + c

Front: RECALL
State how to choose u when integrating by parts.
Back: Pick the u that gets simpler when you differentiate it. A power of x beats a trig or exponential factor, and ln x is always u because there is nothing else you can integrate it as. Choose the other way round and the second integral is worse than the first.