不定積分法及其應用：练习题

Evaluate the indefinite integral: $$\int \left( 5x^4 - \frac{3}{x^2} \right) dx$$

Evaluate the indefinite integral: \( \int \frac{\ln x}{x^2} \, dx \).

Evaluate the indefinite integral: \( \int x^3 e^{x^2} \, dx \).

Evaluate the indefinite integral:$$ \int \frac{e^x}{1+e^{2x}} \, dx $$

Evaluate the indefinite integral:$$ \int \sin(\ln x) \, dx $$

Find the indefinite integral of the function \(f(x) = 5x^4 + 2\sin x\) with respect to \(x\).

Evaluate \(\int x e^{2x} \, dx\) using integration by parts.

Evaluate the indefinite integral: $$ \int x \arctan x \, dx $$

The marginal cost of producing \(x\) units of a product is given by \(C'(x) = 6x^2 - 2e^x\). If the fixed cost (cost when \(x=0\)) is \( \$100 \), find the total cost function \(C(x)\).

Evaluate the indefinite integral \( \int \arcsin x \, dx \).