Integrating Standard Functions
Bayani Gaba-gaba
Integration is differentiation in reverse: it recovers a function from its rate of change, and it measures areas under curves. As with derivatives, a short list of standard integrals does most of the work, with a neat adjustment for the linear expressions \(ax+b\) that appear inside them.
In this lesson you will integrate powers, sines, cosines and the exponential, including the cases \((ax+b)^n\), \(\sin(ax+b)\), \(\cos(ax+b)\) and \(e^{ax+b}\). You will always add the constant of integration \(+c\) for an indefinite integral, the detail examiners look for first.
Manufura
- Integrate functions of the form (ax + b) to the power n (for any rational n, including n = -1), sin(ax + b), cos(ax + b), sec^2(ax + b) and e to the power (ax + b).
- Include an arbitrary constant of integration for indefinite integrals.
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- What is the integral of x^4 with respect to x? A. 4x^3 + c B. x^5/5 + c C. x^5 + c D. x^3/3 + c Answer: B
- What is the integral of (2x + 1)^3 with respect to x? A. (2x + 1)^4/4 + c B. (2x + 1)^4/8 + c C. 3(2x + 1)^2 + c D. (2x + 1)^4 + c Answer: B
- What is the integral of cos(2x)? A. sin(2x) + c B. (1/2) sin(2x) + c C. -(1/2) sin(2x) + c D. 2 sin(2x) + c Answer: B
- What is the integral of e^{3x}? A. e^{3x} + c B. 3e^{3x} + c C. (1/3) e^{3x} + c D. e^{3x}/x + c Answer: C
- For an indefinite integral, you must always: A. divide by x B. add the constant + c C. multiply by 2 D. take a logarithm Answer: B
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