(1) Polynomial and application of the power rule

Determine the indefinite integral of the function
\( f(x) = 3 x^{4} - 1 x^{6} + 3 \).
Apply the linearity of the integral and the power rule for integration. Write the result in the simplest form including the constant of integration.

(2) Model of velocity of motion

The velocity of a body at moment \(x\) is given by the function
\( v(x) = 5 x^{1} + 4 x \).
Determine the position function \(s(x)\) knowing that \(s'(x)=v(x)\). Express the result with the constant of integration and interpret its meaning.

(3) Exponential growth

Calculate the indefinite integral of the function
\( f(x) = 5 e^{6 x} + 3 \).
Simplify the result and explain how to integrate a function of the form \(e^{ax}\).

(4) Trigonometric combination

Determine the indefinite integral of the expression
\( \int \left( 1 \sin(4 x) - 3 \cos(2 x) \right) dx \).
Apply the known formulas for integrals of sine and cosine functions and write the result including the constant of integration.

(5) Logarithmic structure

Calculate the indefinite integral
\( \int \left( \frac{1}{x} + 6 x^{3} \right) dx \).
Express the result using the natural logarithm where necessary and include the constant of integration.

(6) Substitution – power of a linear expression

Determine the indefinite integral
\( \int ( 1 x + 5 )^{4} dx \).
Apply an appropriate substitution and clearly present the final result including the constant of integration.

(7) Roots and powers

Calculate the indefinite integral of the function
\( f(x) = 2 \sqrt{x} + 4 x^{1} - 3 \).
Before integrating, write the root as a power and then apply the power rule.

(8) Physical interpretation of force

The force acting on a body is described by the function
\( F(x) = 1 x^{2} \).
If work is defined as the indefinite integral of force with respect to displacement, determine the work function \(W(x)\). Express the result with the constant of integration.

(9) Fractional expression with several terms

Determine the indefinite integral
\( \int \left( \frac{2}{x} + 1 x^{5} + 4 e^x \right) dx \).
Apply the linearity of the integral and simplify the final expression including the constant of integration.

(10) Tangent and linear term

Calculate the indefinite integral of the expression
\( \int \left( 1 \tan(2 x) + 3 x \right) dx \).
Express the result using the natural logarithm where necessary and add the constant of integration.