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Antiderivatives

This is the opposite of the derivative - and it's an integral part of Calculus. It's uses range from basic integrals to differential equations, with applications in Physics, Chemistry, and Economics.

Level 2

\[ \large \int \sin (x) \, dx - \int \sin (x ) \, dx = \ ? \]

Determine the indefinite integral of the following expression: \[ (\tan{x}) ( \tan{x}+ \sec{x} ).\]

Details and Assumptions:

Use \(C\) as the constant of the integration.

If \(f(x)=\displaystyle \int f(x)dx\),

then \(\displaystyle \int \left( f''(x)+(f'(x))^2+(f(x))^3 \right)dx=0\) will give \[\displaystyle \sum_{r=1}^3 \dfrac{(f(x))^r}{r}=?\]

If \( f\left( \frac{3x-4}{3x+4} \right) = x + 2 \), then what is the antiderivative of \(f(x) \)?

Clarification: Take \(C\) as an arbitrary constant.

If an antiderivative of \(f\left( x \right) \) is \({ e }^{ x }\) and that of \(g\left( x \right) \) is \(\cos { x } \), then \(\displaystyle \int { f\left( x \right) \cos { x } \ dx } +\int { g\left( x \right) { e }^{ x } \ dx } \) is equal to

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