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Antiderivatives: Level 2 Challenges


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

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

Details and Assumptions:

Use CC as the constant of the integration.

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

then (f(x)+(f(x))2+(f(x))3)dx=0\displaystyle \int \left( f''(x)+(f'(x))^2+(f(x))^3 \right)dx=0 will give r=13(f(x))rr=?\displaystyle \sum_{r=1}^3 \dfrac{(f(x))^r}{r}=?

If f(3x43x+4)=x+2 f\left( \frac{3x-4}{3x+4} \right) = x + 2 , then what is the antiderivative of f(x)f(x) ?

Clarification: Take CC as an arbitrary constant.

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


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