Evaluate the integrals in Exercises 91–102.
99. ∫1/(√x (x+1)((arctan√x)²+9)) dx

Guida verificata passo dopo passo

Start by identifying a substitution that simplifies the integral. Notice the presence of \( \sqrt{x} \) inside the integral and in the arctan function. Let \( t = \sqrt{x} \), which implies \( x = t^2 \).

Compute the differential \( dx \) in terms of \( dt \). Since \( x = t^2 \), then \( dx = 2t \, dt \).

Rewrite the integral in terms of \( t \). Substitute \( \sqrt{x} = t \), \( x + 1 = t^2 + 1 \), and \( (\arctan \sqrt{x})^2 + 9 = (\arctan t)^2 + 9 \). Also replace \( dx \) with \( 2t \, dt \).

Simplify the integral expression after substitution. Notice that the \( \sqrt{x} = t \) in the denominator and the \( 2t \, dt \) in the numerator will allow some terms to cancel out, making the integral easier to handle.

Consider a second substitution to handle the \( (\arctan t)^2 + 9 \) term. Let \( u = \arctan t \), then \( du = \frac{1}{1 + t^2} dt \). Use this to rewrite the integral entirely in terms of \( u \) and \( du \), which should simplify the integral to a rational function in \( u \).

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Integration Techniques involving Substitution

Substitution is a method used to simplify integrals by changing variables, often to transform complicated expressions into more manageable forms. In this problem, substituting u = √x can simplify the square root and arctan terms, making the integral easier to evaluate.

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Integrals Involving Natural Logs: Substitution

Properties and Derivatives of the Arctangent Function

Understanding the arctangent function and its derivative is crucial, as the integral contains (arctan(√x))². The derivative of arctan(u) is 1/(1+u²), which helps in recognizing patterns and applying substitution or integration by parts effectively.

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Percorso guidato

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Properties of Functions

Integration involving Rational Functions and Composite Expressions

The integral includes a rational function with terms like 1/(√x (x+1)) and a composite function in the denominator. Recognizing how to handle products and compositions of functions, possibly through partial fractions or algebraic manipulation, is essential to simplify and solve the integral.

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Integrals Involving Natural Logs: Substitution

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