×
Back to all chapters

# 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.

# Antiderivatives: Level 2 Challenges

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

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

×