Find the indefinite integral by substitution method or state that it cannot be found by our...

Question:

Find the indefinite integral by substitution method or state that it cannot be found by our substitution method.

{eq}\displaystyle \int \dfrac {dx} {3 + 2 x} {/eq}.

Integral Formulas:

As we know the power formula of the integration of a function {eq}x^n {/eq} with respect to x where n is a real number that is given by: {eq}\displaystyle \int x^n \ dx = \frac{x^{n +1 }}{n + 1} + C \ \ {/eq} for all real values of n except n = - 1.

When n = -1, then the formula for the integration is: {eq}\displaystyle \int \frac{1}{x} \ dx = \ln |x| + C \ \ {/eq} where C is an arbitrary constant.

Answer and Explanation:


Given:

{eq}\displaystyle \int \frac{1}{3 + 2 x} \ dx {/eq}

Substitute {eq}3 + 2 x = t {/eq}

Differentiate both sides: {eq}\displaystyle 2 \ dx = dt \Rightarrow dx = \frac{dt}{2} {/eq}

Plug in the values of {eq}3 + 2 x {/eq} and {eq}dx {/eq} into the integrand

{eq}\begin{align*} \displaystyle \int \frac{1}{3 + 2 x} \ dx &= \int \frac{1}{t} \times \frac{dt}{2} \\ &= \frac{1}{2} \int \frac{1}{t} \ dt \\ &= \frac{1}{2} \times \ln |t| + C &\text{(Integrating using the formula } \int \frac{1}{x} \ dx = \ln |x| + C \text{)}\\ &= \frac{\ln |t|}{2} + C \\ &\boxed{= \frac{\ln |3 + 2 x|}{2} + C }&\text{(Plugging back in the value of } t \text{)}\\ \end{align*} {/eq}


Learn more about this topic:

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How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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