Determining Which Integration Technique to Use

Calculus 2

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When looking at the integrand, ask yourself the following:

Is it a memorization rule, can it be rewritten, or is the derivative in the problem?
Note: A rewrite might involve an algebraic manipulation (+1 -1 technique, multiplying by a conjugate, completing the square), using an identity, and more.
Try U-Substitution.
Is it a rational function whose denominator can be factored?
Note: If the rational function is improper, you should perform long division.
Try Partial Fraction Decomposition.
Is it a product of two (usually of different type) functions?
Note: A quotient can be turned into a product.
Try Integration by Parts.
Is it a product of sines and cosines, secants and tangents, or cosecants and cotangents?
Try Trigonometric Integration.
Is it in the form of $a^2 - b^2x^2$ , $b^2x^2 - a^2$ , or $a^2 + b^2x^2$ ?
Note: A square root of one of those forms is an ideal case for this technique.
Try Trigonometric Substitution.
If you can't seem to identify which technique would work best for the given integrand, try until you succeed.

Determine which technique to use to compute $\int \tan x \ dx$ . Now, compute it.

Determine which technique to use to compute $\int_1^3 x^4 \ln x \ dx$ . Now, compute it.

Determine which technique to use to compute $\int \frac{1}{(1-x^2)^\frac{3}{2}} \ dx$ . Now, compute it.

Determine which technique to use to compute $\int \sin^3 x \cos x \ dx$ . Now, compute it.

Determine which technique to use to compute $\int \frac{3x^2-2}{x^3-2x-8} \ dx$ . Now, compute it.

Determine which technique to use to compute $\int \frac{3x^2-2}{x^2-2x-8} \ dx$ . Now, compute it.

Determine which technique to use to compute $\int x^3 e^{-2x} \ dx$ . Now, compute it.

Determine which technique to use to compute $\int_0^\frac{\pi}{4} \tan^5 x \sec^3 x \ dx$ . Now, compute it.

Determine which technique to use to compute $\int \frac{1}{\sqrt{25-x^2}} \ dx$ . Now, compute it.

Determine which technique to use to compute $\int \frac{1}{x^2-9} \ dx$ . Now, compute it.

Determine which technique to use to compute $\int \frac{\ln x}{\sqrt{x}} \ dx$ . Now, compute it.

Determine which technique to use to compute $\int \ln x \ dx$ . Now, compute it.