Inverse Trigonometric Functions (AQA A-Level Further Maths): Flashcards

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Inverse Trigonometric Functions
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Reciprocal rule for derivatives

dydx=1dxdy\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}

d(arcsin⁡x)dx\frac{d(\arcsin x)}{dx}

11−x2\frac{1}{\sqrt{1-x^2}}

d(arccos⁡x)dx\frac{d(\arccos x)}{dx}

−11−x2-\frac{1}{\sqrt{1-x^2}}

d(arctan⁡x)dx\frac{d(\arctan x)}{dx}

11+x2\frac{1}{1+x^2}

Sign difference: derivatives of arcsin⁡x\arcsin x and arccos⁡x\arccos x

Differ only by a negative sign

∫11−x2dx\int \frac{1}{\sqrt{1-x^2}} dx

arcsin⁡x+c\arcsin x + c

∫11+x2dx\int \frac{1}{1+x^2} dx

arctan⁡x+c\arctan x + c

Substitution for integrals with a2−x2\sqrt{a^2 - x^2}

x=asin⁡ux = a\sin u

Substitution for integrals with a2+x2a^2 + x^2

x=atan⁡ux = a\tan u

When to complete the square in these integrals

When denominator is quadratic not matching standard form

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