Practise calculus with a clear purpose

1. 1
   Factor

   Use the difference of squares.

   $$x^2-9=(x-3)(x+3)$$
2. 2
   Simplify near 3

   Cancel the common factor for x not equal to 3.

   $$\frac{(x-3)(x+3)}{x-3}=x+3$$
3. 3
   Take the limit

   The simplified expression is continuous at 3.

   $$\lim_{x\to3}(x+3)=6$$

Independent check: Values just below and above 3 approach 6.

1. 1
   Recognize the product

   Use the product rule.

   $$(uv)'=u'v+uv'$$
2. 2
   Differentiate the factors

   Differentiate x squared and sine x.

   $$(x^2)'=2x,\quad(\sin x)'=\cos x$$
3. 3
   Combine

   Keep the unchanged factor in each term.

   $$f'(x)=2x\sin x+x^2\cos x$$

Independent check: Central difference slopes agree at regular test points.

1. 1
   Antidifferentiate

   Integrate each power term.

   $$\int(3x^2-2x)\,dx=x^3-x^2$$
2. 2
   Evaluate

   Use upper endpoint minus lower endpoint.

   $$[x^3-x^2]_0^2=(8-4)-0=4$$

Independent check: Numerical quadrature on the original integrand also gives 4.

1. 1
   Choose the parts

   Differentiate the logarithm and integrate x.

   $$u=\ln x,\quad dv=x\,dx$$
2. 2
   Confirm progress

   The remaining integral is elementary.

   $$\frac{x^2}{2}\ln x-\int\frac{x}{2}\,dx$$

Independent check: Differentiate the antiderivative on x greater than zero.

1. 1
   Apply Part 1

   Evaluate the integrand at x squared.

   $$\cos(x^2)$$
2. 2
   Apply the chain rule

   Multiply by the derivative of the upper bound.

   $$F'(x)=\cos(x^2)(2x)$$

Independent check: Differentiate F(x) = sin(x²) - sin(1).

1. 1
   Determine the sign

   To the left of 2, x minus 2 is negative.

   $$x-2\to0^-$$
2. 2
   Take the reciprocal

   Magnitude grows while the sign stays negative.

   $$\frac1{x-2}\to-\infty$$

Independent check: For x = 2 - 1/n, the function value is -n.

1. 1
   Name the layers

   Let u be the inner expression.

   $$u=3x^2-1,\quad f=u^4$$
2. 2
   Chain the derivatives

   Multiply the outer derivative by the inner derivative.

   $$f'(x)=4(3x^2-1)^3(6x)$$

Independent check: At x = 1, a centered difference quotient approaches 192.

1. 1
   Split the terms

   Use linearity.

   $$6\int x^2dx-4\int\frac1x dx$$
2. 2
   Integrate

   Use the power and logarithmic rules.

   $$2x^3-4\ln|x|+C$$

Independent check: Differentiation recovers the integrand on either side of zero.

1. 1
   Substitute

   Choose the inner expression.

   $$u=x^2+1,\quad du=2x\,dx$$
2. 2
   Integrate and return

   Integrate cosine, then substitute back.

   $$\sin u+C=\sin(x^2+1)+C$$

Independent check: The chain rule differentiates the result to 2x cos(x² + 1).

1. 1
   Choose parts

   Differentiate x and integrate the exponential.

   $$u=x,\quad v=\frac12e^{2x}$$
2. 2
   Apply and simplify

   Use integration by parts.

   $$\frac{x}{2}e^{2x}-\frac14e^{2x}+C$$

Independent check: The product rule returns x e^(2x).