Trigonometric Identities and Integration Techniques

1/(x^2 + 8x + 1)

1/(x^2 - 8x + 1)

1/(x^2 - 8x - 1)

1/(x^2 + 8x - 1)

Partial fraction decomposition

Substitution method

Integration by parts

Completing the square

To simplify the numerator

To factor the quadratic expression

To transform the quadratic into a perfect square form

To eliminate the constant term

16

8

15

4

Integrate directly

Differentiate the expression

Apply trigonometric substitution

Use partial fraction decomposition

Sine

Secant

Cosine

Tangent

sin^2(theta) + cos^2(theta) = 1

1 + tan^2(theta) = sec^2(theta)

1 - sin^2(theta) = cos^2(theta)

tan^2(theta) - 1 = sec^2(theta)

15 * tan^2(theta)

15 * sec^2(theta)

15 * cos^2(theta)

15 * sin^2(theta)

Use a trigonometric identity

Apply another substitution

Draw a triangle and use Pythagorean theorem

Differentiate the result

Secant is the difference between cosine and sine

Secant is the square of cosine

Secant is the sum of cosine and sine

Secant is the reciprocal of cosine