Find the following indefinite integrals: (a) ∫(4+x^2)^-1 d x ; (b) ∫(8+2 x-x^2)^-1 / 2 d x for

step 1 Okay, so we need to find the indefinite integrals. The first integral is the integral of 1 over

Find the following indefinite integrals: (a) ∫(4+x^2)^-1 d x...

Key Concepts

Standard Integral Formulas

Many integrals can be evaluated by recognizing and applying standard formulas. In particular, integrals that have a form resembling 1/(a^2 + x^2) are directly handled using the arctan formula. Knowing these basic forms allows one to quickly identify antiderivatives for common rational functions without needing lengthy derivations.

Completing the Square

Completing the square is a crucial algebraic technique that transforms quadratic expressions into a form that is more suitable for integration. By rewriting a quadratic polynomial in a perfect square plus or minus a constant, one can reduce the integral to a standard form, typically leading to inverse trigonometric functions such as the arcsine or arctangent. This method is especially useful when dealing with quadratic expressions in the denominator of an integrand.

Trigonometric Substitution

Trigonometric substitution is an integration method used to handle integrals involving radical expressions, often resulting from quadratic forms. By substituting a trigonometric function for a variable, the complicated algebraic structure is transformed into a simpler trigonometric integral. This substitution leverages trigonometric identities to simplify the square root or squared expressions, making the integration process more straightforward.

Trigonometric Identities and Half-Angle Formulas

Trigonometric identities, such as the half-angle formulas, are instrumental in simplifying integrals involving trigonometric functions. These identities help transform sums or differences of trigonometric functions into a product or a single trigonometric function, thereby reducing the complexity of the integrand. This technique is useful in integrals where functions like sin or cos appear inside a rational function.

Substitution Method

The substitution method is a fundamental tool in integration that involves replacing a part of the integrand with a new variable. This technique simplifies the integral by regrouping its parts, often transforming it into a form that can be directly integrated using standard formulas. It is particularly effective when the derivative of the substituted expression is present elsewhere in the integrand, thus facilitating the integration process.

Transcript

00:01 Okay, so we need to find the indefinite integrals.

00:04 The first integral is the integral of 1 over 4 plus x squared dx.

00:14 So if we rewrite this is the integral of 1 over 2 squared plus x squared dx, then it's in the form that we're used to for inverse tangent.

00:25 So this is 1 half, the inverse tangent of x over 2.

00:30 Do not forget your plus c or b we have the integral of one over the square root of 8 plus 2x minus x squared the x we can factor out that negative sign and then complete the square so this becomes the integral of 1 over the square root of negative x minus 1 squared minus 3 squared.

01:17 So that is adding and subtracting 1 squared to help us complete the square here.

01:27 And this is in the form of inverse sign.

01:31 So this is the inverse sign of x minus 1 over 3 plus c.

01:46 C, we have the integral of 1 over 1 plus sine of theta d theta, which is equal to the integral of 1 minus sine of theta over, so we're making the difference of two squares by multiplying and dividing by 1 minus sine of theta.

02:09 So 1 plus sine of theta, 1 minus sine of theta, d theta.

02:19 This is the integral of 1 minus sine of theta over 1 minus sine squared of theta, which d theta, which makes it the integral of 1 minus sine of theta over cosine squared of theta.

02:37 And i can split this up into two integrals.

02:40 The first integral is 1 over cosine squared of theta, which would be the integral of secan squared of theta, d theta, and then sine over cosine square...

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