Integration techniques

5 ECAT Mathematics questions with answers and explanations.

All questions with answers

What is the value of the integral of (2x + 3) dx from 0 to 1?
- 1 3
- 2 4
- 3 5
- 4 2
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Correct answer: 4

The antiderivative is x^2 + 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 3 e^(3x) + C
- 2 e^(3x) / 3 + C
- 3 e^(3x) + C
- 4 e^(x) / 3 + 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 + x^2) dx?
- 1 ln(1 + x^2) + C
- 2 tan inverse of x + C
- 3 1 / x + C
- 4 sin 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 + x^2).

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What is the area enclosed between y = x and y = x^2 from x = 0 to x = 1?
- 1 1/2
- 2 1/3
- 3 1/6
- 4 1/4
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Correct answer: 1/6

The area is the integral of (x - x^2) from 0 to 1, which is 1/2 - 1/3 = 1/6.

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What is the integral of 2x / (x^2 + 1) dx?
- 1 ln(x^2 + 1) + C
- 2 x^2 / (x^2 + 1) + C
- 3 2 ln x + C
- 4 1 / (x^2 + 1) + C
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Correct answer: ln(x^2 + 1) + C

The numerator is the derivative of the denominator, so the integral is ln(x^2 + 1) + C.

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