Applications of Integration

U-substitution

Partial fractions after polynomial long division

Trigonometric substitution

Integration by parts

Find $\int_{{-\pi}/{4}}^{10}(t^2+3)^{-\frac{1}{2}}dt$

Compute $\int_{{4\pi}/5}^{20\pi}(t^2+3)^{\frac{1}{2}}dt$

Evaluate $\int_{{-\infty}}^{\infty} (t^2+3) dt$

Evaluate $\int_{{0}}^{{4\pi}} \sqrt{(t^2+3)} dt$

A new computation method for arc length is needed as addition alters fundamental properties of integrands involved in arc lengths.

The arc length decreases as it effectively lowers gradient magnitudes integrated along [a,b].

There is no effect on the arc length because adding a constant shifts the graph vertically without changing its shape.

The arc length increases due to an increase in values of $h(x)$ across [a,b].

3

4

5

2

$\int_{0}^{\sqrt{\pi}} \sin(x) dx$

$\int_{0}^{\sqrt{\pi}} x\sin(x) dx$

$\int_{0}^{\sqrt{\pi}} \cos(x) dx$

$\int_{0}^{\sqrt{\pi}} \sqrt{1 + 4x^2\cos^2(x^2)} dx$

$A = - \frac{1}{4} \sin(4\theta) \bigg|_0^\pi$

$A = - \frac{1}{6} \sin(3\theta) \bigg|_0^\pi$

$A = \frac{1}{4} \int_0^\pi \cos(4\theta) d\theta$

$A = \frac{1}{2} \int_0^\pi \cos^2(2\theta) d\theta$

They provide a visual representation of slopes at various points which helps predict behavior of solutions without solving them explicitly.

Slope fields are mainly tools for calculating area under curves represented by differential equations' solutions.

They offer numerical data which can be used directly in formulas for solving differential equations analytically.

Slope fields are used solely for determining equilibrium solutions where slopes are zero across the field.

Applying Rectangular Approximation regardless of interval count due to its simplicity and speed.

Employing Midpoint Rule with few intervals assuming linear behavior between midpoints.

Averaging multiple Trapezoidal approximations made with random subinterval counts.

Using Simpson's Rule with an adequate number of intervals depending on curvature variability.

Integrate within absolute values the velocity in respective intervals

Multiply the average speed by the duration of movement in each segment

Take the difference between the final and initial velocities

Integrate over the entire range without considering changes in direction

Trigonometric substitution

Partial fraction decomposition

U-substitution

Integration by parts