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Integration techniques
5 ECAT Mathematics questions with answers and explanations.
Printable worksheet with answer key
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What is the value of the integral of (2x + 3) dx from 0 to 1?
- 1 4
- 2 5
- 3 2
- 4 3
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Correct answer: 4
The antiderivative is x² + 3x. At 1 it is 4, and at 0 it is 0, so the value is 4.
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What is the integral of e^(3x) dx?
- 1 e^(x) / 3 + C
- 2 e^(3x) + C
- 3 e^(3x) / 3 + C
- 4 3 e^(3x) + C
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Correct answer: e^(3x) / 3 + C
Dividing by the derivative of the exponent, 3, gives e^(3x) / 3 + C.
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What is the integral of 1 / (1 + x²) dx?
- 1 ln(1 + x²) + C
- 2 1 / x + C
- 3 sin inverse of x + C
- 4 tan inverse of x + C
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Correct answer: tan inverse of x + C
The derivative of the inverse tangent of x is 1 / (1 + x²).
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What is the area enclosed between y = x and y = x² from x = 0 to x = 1?
- 1 1/3
- 2 1/2
- 3 1/6
- 4 1/4
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Correct answer: 1/6
The area is the integral of (x - x²) from 0 to 1, which is 1/2 - 1/3 = 1/6.
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What is the integral of 2x / (x² + 1) dx?
- 1 ln(x² + 1) + C
- 2 x² / (x² + 1) + C
- 3 1 / (x² + 1) + C
- 4 2 ln x + C
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Correct answer: ln(x² + 1) + C
The numerator is the derivative of the denominator, so the integral is ln(x² + 1) + C.