Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(t√t + √t) / t² dt

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First, simplify the integrand by expressing all terms with exponents. Recall that \( \sqrt{t} = t^{1/2} \) and rewrite the expression inside the integral accordingly.

Rewrite the integrand \( \frac{t\sqrt{t} + \sqrt{t}}{t^2} \) as \( \frac{t \cdot t^{1/2} + t^{1/2}}{t^2} = \frac{t^{3/2} + t^{1/2}}{t^2} \).

Split the fraction into two separate terms: \( \frac{t^{3/2}}{t^2} + \frac{t^{1/2}}{t^2} \). Then simplify each term by subtracting exponents in the denominator from those in the numerator.

After simplification, express the integrand as a sum of powers of \( t \): \( t^{3/2 - 2} + t^{1/2 - 2} = t^{-1/2} + t^{-3/2} \).

Now, integrate each term separately using the power rule for integration: \( \int t^n dt = \frac{t^{n+1}}{n+1} + C \), making sure to add the constant of integration at the end.

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Indefinite Integral and Antiderivative

An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. Understanding this helps in solving integrals without specified limits.

Algebraic Simplification of the Integrand

Before integrating, simplify the integrand by combining like terms and rewriting expressions using exponent rules. For example, rewrite roots as fractional exponents and simplify fractions to make integration straightforward and reduce errors.

Power Rule for Integration

The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is fundamental for integrating polynomial and power functions, allowing direct computation once the integrand is expressed in terms of powers.

Finding Indefinite IntegralsIn Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.∫(t√t + √t) / t² dt