Review of Basic Integration and Substitution

This lesson reviews the integration tools from Calculus 1 to make sure your foundation is solid before we add new techniques.

Power rule:

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$

Special cases:

$$\int \frac{1}{x} \, dx = \ln|x| + C$$ $$\int e^x \, dx = e^x + C$$

Trig:

$$\int \sin x \, dx = -\cos x + C \qquad \int \cos x \, dx = \sin x + C$$

Constants factor out, and sums/differences split term by term. These rules handle any polynomial, basic trig, and exponential integral directly.

When the integrand contains a composite function and the derivative of the inside is present (or close to it), substitution simplifies the integral.

1. Choose u = the inside function
2. Compute du
3. Rewrite entirely in terms of u and du
4. Integrate
5. Substitute back to x

For a general exponential:

$$\int e^{kx}\,dx = \frac{1}{k}e^{kx} + C$$

This comes from substituting u = kx, du = k dx.

Example 1: Polynomial

$$\int (4x^3 - 6x + 7)\,dx = x^4 - 3x^2 + 7x + C$$

Each term uses the power rule independently.

Example 2: Basic substitution

Evaluate:

$$\int 2x(x^2 + 5)^3\,dx$$

Let u = x² + 5, du = 2x dx.

$$\int u^3\,du = \frac{u^4}{4} + C = \frac{(x^2 + 5)^4}{4} + C$$

Example 3: Trig with substitution

Evaluate:

$$\int \sin(4x)\,dx$$

Let u = 4x, du = 4 dx, so dx = du/4.

$$\frac{1}{4}\int \sin u\,du = -\frac{1}{4}\cos u + C = -\frac{1}{4}\cos(4x) + C$$

Example 4: Square root substitution

Evaluate:

$$\int x\sqrt{x^2 - 1}\,dx$$

Let u = x² - 1, du = 2x dx, so x dx = du/2.

$$\frac{1}{2}\int \sqrt{u}\,du = \frac{1}{2}\cdot\frac{2}{3}u^{3/2} + C = \frac{1}{3}(x^2 - 1)^{3/2} + C$$

These are the building blocks for everything in Calculus 2:

• Every advanced technique (integration by parts, partial fractions, trig substitution) eventually reduces to basic rules
• Physics problems start with setting up the integral, then you need these rules to evaluate it
• U-substitution is the most frequently used technique in all of calculus