Integration and Evaluation

00:00 6 marks

7. [Maximum mark: 6]

(a) Find \(\int \left(2x^3 + \frac{3}{x^2} – 5\right) dx\). [3]
(b) Evaluate \(\int_1^e \frac{2}{x} dx\). [3]

Part a

5 minutes 3 marks

Question

Find \(\int \left(2x^3 + \frac{3}{x^2} – 5\right) dx\). [3]

Hint Hint

Hint

Separate the integrals for \(2x^3\), \(\frac{3}{x^2}\), and \(-5\), and apply the power rule for each term.

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Complete Answer

\(\frac{1}{2}x^4 – \frac{3}{x} – 5x + C\)

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Part b

5 minutes 3 marks

Question

Evaluate \(\int_1^e \frac{2}{x} dx\). [3]

Hint Hint

Hint

Find the antiderivative \(2\ln|x|\) and then substitute the limits \(x = e\) and \(x = 1\).

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