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Integration : Key Stage 5 (A2-Level) : Mathematics Content Knowledge


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Key Stage 5 (A2-Level)
Integration
Question 1 of 5

1. How confident are you that you understand and can explain:

a. Integration of e^x, 1/x, sin x and cos x?


Example



Example 1 Example 2
   

\int {e^x dx = e^x  + c}

\int {\frac{1}{x}} dx = \ln x + c

\int {\sin xdx =  - \cos x + c} if x is measured in radians

\int {\cos xdx = \sin x + c} if x is measured in radians

Example:
Find \int {\sin x \ \cos x \ dx}.

By inspection
\frac{d}{{dx}}(\sin ^2 x) = 2\sin x\cos x

So:
\int {\sin x \ \cos x \ dx}  = \frac{1}{2}\sin ^2 x + c

Example:
Find \int {e^{3x} } dx.

By inspection
\frac{d}{{dx}}(e^{3x} ) = 3e^{3x}so \int {e^{3x} } dx = \frac{1}{3}e^{3x}  + c

Example:
Find\int {\frac{{2x}}{{x^2  + 1}}} dx

Let u = x^2  + 1. Then \frac{{du}}{{dx}} = 2x

So \int {\frac{{2x}}{{x^2  + 1}}} dx = \int {\frac{1}{u}} du = \ln u + c

= \ \ln (x^2  + 1) + k

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