# Calculus

## U-Substitution

The u-substitution method replaces part of our integral with a variable so that it is easier to integrate. This can apply if we have large exponents or rational expressions.

Consider $$\int (2t-3)^{29} dt$$. Let us use the u-substitution method to replace $$2t-3$$ with $$u$$.

$$\Longrightarrow$$ differentiating both sides, we obtain $$2 = \frac{du}{dt} \rightarrow dt = \frac{du}{2}$$

$$\Longrightarrow$$ substituting for $$2t-3$$ and $$dt$$: $\int \frac{u^{29}}{2} du$

$\Longrightarrow \ = \frac{1}{2}\frac{u^{30}}{30} + C$

$\ \ \ \ \ \ \ \ \ \ \Longrightarrow \ = \frac{1}{60}(2t-3)^{30} + C$

## Integration by Parts

Integration by parts is a method that used for integrating products of two functions, represented by the following formula: $\int f(x)g'(x)dx = f(x)g(x) - \int f'(x)g(x)dx$

When approaching an integral with this method, the first step is determining which part of our integral is easiest to integrate by itself so that from our $$g'(x)$$ we can figure out our $$g(x)$$ in the formula.

The general rule when choosing $$f(x)$$ is to follow this order:

I nverse Trigonometric

L ogs

A lgebraic

T rigonometric

E xponential

Before continuing onto our example, recall the integrals for our trigonometric functions:

$\int tanxdx = -ln(cosx) + C \ \ \ \ \ \ \ \ \ \ \int cotx dx = ln(sinx) + C$

$\int secx dx = ln(secx + tanx) + C \ \ \ \ \ \ \ \ \ \ \int cscx dx = -ln(cscx + cotx) + C$

Solve the following indefinite integral: $$\int xsin(x) dx$$.

$$\Longrightarrow$$ As per our ILATE structure, we choose $$f(x) = x$$ and $$g '(x) = sin(x)$$

$\Longrightarrow \ \int xsin(x) dx = x(-cos(x)) - \int (1)(-cos(x)) dx$

$\Longrightarrow \ \ \ \ \ \ \ \ \ \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \ = -xcos(x) - (-sin(x)) + C \ \ \ \ \ \ \ \ \$

$\Longrightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = -xcos(x) + sin(x) + C \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$

EXAMPLE

1. Solve for the indefinite integral $$\int_{a}^{b} e^x sin(x) dx$$

2. Solve for the definite integral $$\int_{0}^{1} x e^x dx$$

See the video below for the solution: