Integration and Accumulation of Change

When evaluating definite integrals over closed bounded intervals without singularities.

One where it is not clear if the integral converges because of singular behavior at limits of integration.

Where definite integrals involve discontinuous integrands over fixed intervals.

A theorem that relates differentiation and antidifferentiation.

A theorem that relates integration and differentiation.

A theorem that relates convergence and divergence.

A theorem that relates limits and continuity.

Zero

One

The largest positive value for k within domain constraints

The smallest negative value for k within domain constraints

Integration by parts, as it typically handles products of functions.

Trigonometric substitution, because it simplifies the integrand into a form that can be easily integrated.

Partial fraction decomposition, which is more suitable for rational functions with polynomials in the numerator and denominator.

Direct integration, since there is no elementary antiderivative in its current form.

$f''(x)$

$\int f(x) , dx$

$f'(x)$

$F(x)$

Removing steps elsewhere, not at zero

Increasing the step size from a round number to a rounder number

Changing step locations to be closer together

Adding a step at zero

$x^5$

$20x^3$

$5x^3 + C$

$4x^5$

Partial fractions

Power rule

Integration by parts

Trigonometric substitution

Applying integration by parts directly on $e^{5\sqrt{x}}$ without any preliminary substitutions or rewrites.

Utilizing Taylor series expansion around $x=0$ prior to integration which isn't typically required for this form.

Setting up u-substitution with $u=5\sqrt{x}$ followed by rewriting the exponential expression before integrating.

Simplifying exponentials via Euler's formula before integrating, although this doesn't apply directly here.

Changing initial slope $f'(0)$

Increasing initial condition by a factor of ten

Subtracting five from initial condition $f(0)$

Multiplying initial condition by negative one