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Writing an integral down is only the first step. A toolkit of techniques can help find its value, from substitutions to trigonometry to partial fractions to differentiation.

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Joel Yip**

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\[\large\int_{1}^{a}\dfrac{x-1}{x+\sqrt{x}}dx=4\]

Let \(a>1\) be a constant satisfying the equation above. What is the value of \(\large a^2+a+1\)?

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Ikkyu San**

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Evaluate the following integral

\[\large\displaystyle \int \dfrac{\sin(x)}{\cos^3(x)} \, \Bbb{d}x \]

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Trevor Arashiro**

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\[ \large \int_0^2 x \sqrt{1-(x-1)^2} \, dx = \ ? \]

Give your answer to 2 decimal places.

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Majed Musleh**

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Evaluate the indefinite integral below.

\[\large \int \frac{dx}{1 + e^{-x}}\]

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Andrew Ellinor**

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