Find int(dx)/(ax+b) (1) log_(e)(ax+b)+c (3) C+1//alog_(e)(ax+b)

Identify Integral: Identify the integral to be solved. We need to evaluate the integral of the function 1/(ax+b) with respect to x.Use Substitution: Use a substitution to simplify the integral. Let u = ax + b. Then, du = a dx, which implies dx = du/a.Rewrite in Terms of u: Rewrite the integral in terms of u. The integral becomes int(1/u) * (du/a) = (1/a) * int(1/u) du.Evaluate Integral of 1/u: Evaluate the integral of 1/u. The integral of 1/u with respect to u is ln|u| + C, where C is the constant of integration.Substitute Back: Substitute back the original variable. Since u = ax + b, we have ln|u| + C = ln|ax + b| + C.Multiply by Constant: Multiply by the constant 1/a from the substitution. The final answer is (1/a) * ln|ax + b| + C.

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Find $\int \frac{dx}{ax+b}$ $\newline$ ( $1$ ) $\log_{e}(ax+b)+c$ $\newline$ ( $3$ ) $C+\frac{1}{a}\log_{e}(ax+b)$

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Calculus

Evaluate definite integrals using the chain rule

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Q. Find

\int \frac{dx}{ax+b}

\newline

(

1

)

\log_{e}(ax+b)+c

\newline

(

3

)

C+\frac{1}{a}\log_{e}(ax+b)

Identify Integral: Identify the integral to be solved. $\newline$ We need to evaluate the integral of the function $\frac{1}{ax+b}$ with respect to $x$ .
Use Substitution: Use a substitution to simplify the integral. $\newline$ Let $u = ax + b$ . Then, $du = a dx$ , which implies $dx = \frac{du}{a}$ .
Rewrite in Terms of $u$ : Rewrite the integral in terms of $u$ . The integral becomes $\int(\frac{1}{u}) \cdot (\frac{du}{a}) = (\frac{1}{a}) \cdot \int(\frac{1}{u}) du$ .
Evaluate Integral of $1/u$ : Evaluate the integral of $1/u$ . The integral of $1/u$ with respect to $u$ is $ext{ln}|u| + C$ , where $C$ is the constant of integration.
Substitute Back: Substitute back the original variable. $\newline$ Since $u = ax + b$ , we have $\ln|u| + C = \ln|ax + b| + C$ .
Multiply by Constant: Multiply by the constant $\frac{1}{a}$ from the substitution. $\newline$ The final answer is $\frac{1}{a} \cdot \ln|ax + b| + C$ .