Calculus - The indefinite integral

Find the indefinite integral: $\int (4x^3 - 6x^2 + 5) dx$.

$x^4 - 2x^3 + 5x + C$

Apply the power rule to each term separately: $\int 4x^3 dx = 4 \cdot \frac{x^4}{4} = x^4$; $\int -6x^2 dx = -6 \cdot \frac{x^3}{3} = -2x^3$; $\int 5 dx = 5x$. Finally, add the constant $C$.

Given that $f'(x) = e^{2x} + \frac{3}{x}$ and $f(1) = \frac{1}{2}e^2$, find the expression for $f(x)$.

$f(x) = \frac{1}{2}e^{2x} + 3\ln|x|$

First, find the general integral: $\int (e^{2x} + \frac{3}{x}) dx = \frac{1}{2}e^{2x} + 3\ln|x| + C$. Use the condition $f(1) = \frac{1}{2}e^2$ to find $C$: $\frac{1}{2}e^{2(1)} + 3\ln|1| + C = \frac{1}{2}e^2 \implies \frac{1}{2}e^2 + 0 + C = \frac{1}{2}e^2 \implies C = 0$.

Evaluate $\int \cos(3x - 4) dx$.

$\frac{1}{3}\sin(3x - 4) + C$

This is an integral of the form $\int \cos(ax + b) dx$. Using the rule for linear compositions, we divide by the coefficient of $x$ (which is $3$) and integrate the outer function: $\frac{1}{3} \sin(3x - 4) + C$.

Find $\int \frac{2x^2 + 1}{x} dx$.

$x^2 + \ln|x| + C$

First, simplify the integrand by dividing each term in the numerator by $x$: $\int (\frac{2x^2}{x} + \frac{1}{x}) dx = \int (2x + \frac{1}{x}) dx$. Integrating gives $2 \cdot \frac{x^2}{2} + \ln|x| + C = x^2 + \ln|x| + C$.