Standard Integration Techniques

Assume $g$ is continuously differentiable and $f$ is continuous on an interval containing its image. If $F'=f$, the chain rule gives $(F(g(x)))'=f(g(x))g'(x)$: this is why substitution works. For definite integrals,

$$\int_a^b f(g(x))g'(x)\,dx=\int_{g(a)}^{g(b)}f(u)\,du.$$

Transform both the differential and the bounds; do not combine $u$-integrands with $x$-bounds. This forward formula does not require $g$ to be one-to-one. Solving for $x$ in terms of $u$, however, may require separate inverse branches. For indefinite integrals, substitute back and add $C$.

The trigonometric power shortcuts below assume nonnegative integer powers and that the saved factors are actually present: at least two secants for the even-secant rule, at least one secant and one tangent for the odd-tangent rule. Work on intervals without poles ($\cos x\ne0$ in the tangent/secant examples).

U Substitution

The u substitution method is useful for integrals involving a composite function. It follows the principle that:

$\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du$

where $u = g(x)$ and $du = g'(x)dx$. This technique is also applicable without limits for indefinite integrals.

Example

For the integral $\int_1^2 5x^2 \cos(x^3) \, dx$, we set $u = x^3$ which gives us $du = 3x^2 \, dx$. The limits are also changed accordingly to $u$ values. The integral simplifies to:

$\int_1^8 \frac{5}{3} \cos(u) \, du = \left[ \frac{5}{3} \sin(u) \right]_1^8 = \frac{5}{3} (\sin(8) - \sin(1))$

Work through the definite-integral examples in OpenStax’s substitution chapter, writing the new bounds beside each change of variable. They provide practice checking that the integrand, differential, and bounds all use the same variable.

Products and Quotients of Trig Functions

For Products

When integrating products of sine and cosine, consider the following strategies based on the powers of sine and cosine:

• If the power of sine is odd, move one sine out and convert the rest to cosines using $\sin^2(x) = 1 - \cos^2(x)$.
• If the power of cosine is odd, move one cosine out and convert the rest to sines using $\cos^2(x) = 1 - \sin^2(x)$.
• If both powers are odd, use the above strategies.
• If both powers are even, use double angle or half angle formulas.

For Quotients

When integrating products of tangent and secant, the strategies are as follows:

• If the power of secant is even, save a factor of $\sec^2(x)$, convert the remaining even powers of secant with $\sec^2(x)=1+\tan^2(x)$, and use $u=\tan(x)$.
• If the power of tangent is odd, save a factor of $\sec(x)\tan(x)$, convert the remaining even powers of tangent with $\tan^2(x)=\sec^2(x)-1$, and use $u=\sec(x)$.
• Other exponent combinations may require identities or a different rearrangement rather than either shortcut.

Example - Products

For the integral $\int \tan^3(x) \sec^5(x) \, dx$, use $u = \sec(x)$, $du = \sec(x)\tan(x) \, dx$. This leads to:

$\int u^4(u^2 - 1) \, du = \int (u^6 - u^4) \, du$

After integration, revert back to $x$:

$\frac{1}{7} \sec^7(x) - \frac{1}{5} \sec^5(x) + C$

Example - Quotients

For the integral $\int \frac{\sin^3(x)}{\cos^3(x)} \, dx$, use the identity $\sin^2(x) = 1 - \cos^2(x)$ and the substitution $u = \cos(x)$:

Because $du=-\sin(x)\,dx$,

$$\int \frac{(1 - \cos^2(x))\sin(x)}{\cos^3(x)} \, dx = -\int \left(u^{-3}-u^{-1}\right)\,du.$$

After integrating and substituting $u=\cos(x)$:

$\frac{1}{2}\sec^2(x) + \ln|\cos(x)| + C$

Trig Formulas

The trigonometric formulas useful in these integrations include:

$\sin(2x) = 2 \sin(x) \cos(x)$ $\cos(2x) = 1 - 2 \sin^2(x)$ $\sin^2(x) = \frac{1}{2}(1 - \cos(2x))$

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