Standard Integration Techniques

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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:

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

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 tangent is odd, save one tangent and convert the rest to secants.
• If the power of secant is even, save one secant and convert the rest to tangents.
• Use appropriate substitutions such as $u = \tan(x)$ or $u = \sec(x)$ depending on the scenario.

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 $u = \cos(x)$:

$\int \frac{(1 - \cos^2(x))\sin(x)}{\cos^3(x)} \, dx = \int \left( \frac{1}{u^2} - \frac{1}{u^4} \right) \, du$

After integrating and reverting back to $x$:

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

Trig Formulas

The trigonometric formulas useful in these integrations include:

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