Integration and Accumulation of Change

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

Zero

One

The largest positive value for k within domain constraints

The smallest negative value for k within domain constraints

$\int_0^a f(x) , dx$

$\sum_{x=a}^{b} f(x)\Delta x$

$\int_a^b |f(x)| , dx$

$\int_a^b f(x) , dx$

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.

Utilizing implicit differentiation until an explicit function is derived.

Differentiating under the integral sign using Leibniz's rule.

Applying L'Hôpital's Rule till an integrable form appears.

Numerical integration methods such as Simpson's rule or trapezoidal rule.

Trigonometric substitution

U-substitution

Integration by parts

Partial fraction decomposition

Partial fractions

Power rule

Integration by parts

Trigonometric substitution

$f''(x)$

$\int f(x) , dx$

$f'(x)$

$F(x)$

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.

Doubling $a$

Subtracting 3 from $b$

Tripling $c$

Adding 5 to $b$