Integration and Accumulation of Change

$x \cdot \cos(x) + \sin(x) + C$

$x \cdot \cos(x) - \cos(x) + C$

$x \cdot \cos(x) + \cos(x) + C$

$-x \cdot \cos(x) + \sin(x) + C$

When the integrand is a polynomial function.

When the integrand contains logarithmic functions.

When the integrand contains trigonometric functions.

When the integrand is a product of two functions

$e^{x}dx$

$e^{3}dx$

$xe^{3}dx$

$e^{3x}dx$

Integrate $dv$ to find $v$.

Choose functions $u$ and $dv$ from the integrand.

Solve for $du$.

Differentiate $u$ to find $du$.

$\frac{e}{(17 \times \frac{1}{7})}$

$-7t e^{7t}$

$-t e^{- \frac{7}{17}}$

$- \frac{7}{e^{77}} \times \frac{1}{7}$

$\int_a^b 3x^2+3$

$\int_a^b 2x+5$

$\int_a^b 30x^3+20x^2$

$\int_a^b e^x \cos(x)$

$\sin(3x)$

$e^{2x}$

$2x$

$3x$

$\int_a^b f(x)g(x) , dx = \int_a^b f'(x)g'(x) , dx$

$\int_a^b f(x)g'(x) , dx = \left. (f(x)g(x)) \right|_a^b - \int_a^b f'(x)g(x) , dx$

$\int_a^b f(x)g'(x) , dx = \left. (f(x)g(x)) \right|_a^b \cdot \int_a^b f'(x)g(x) , dx$

$\int_a^b f(x)g'(x) , dx = \left. (f(x)g(x)) \right|_a^b + \int_a^b f'(x)g(x) , dx$

The exponential function regardless of other terms.

The function that complicates when differentiated.

The constant function, if present.

The function that simplifies when differentiated.

$\frac{1}{2} \cdot x \cdot e^{2x} - \frac{1}{4} \cdot e^{2x} + C$

$\frac{1}{2} \cdot x \cdot e^{2x} - e^{2x} + C$

$\frac{1}{2} \cdot x \cdot e^{2x} - \frac{1}{2} \cdot e^{2x} + C$

$\frac{1}{2} \cdot x \cdot e^{2x} - \frac{1}{4} \cdot e^{4x} + C$