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

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Key Stage 5 (A2-Level)
Integration
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# 1. How confident are you that you understand and can explain:

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

 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 $\int {\frac{{2x}}{{x^2 + 1}}} dx = \int {\frac{1}{u}} du = \ln u + c$ $= \ \ln (x^2 + 1) + k$