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Integration Techniques

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.

Level 2

         

\[\large \int_{-\frac \pi 2 }^{\frac \pi 2 } \bigg [ e^{ \sin(x)} \cos (x )\bigg ] \ dx = a - \frac 1 a, \ \ \ a > 0, \ \ \ \ \ a = \ ? \]

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

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

Evaluate the following integral

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

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

Give your answer to 2 decimal places.

Evaluate the indefinite integral below.

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

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