Question

1. find each indefinite integral. (a) $\int 7 dx$ (b) $\int 9x^2 dx$ (c) $\int \frac{1}{x} dx$ (d) $\int \frac{1}{x^4} dx$ (e) $\int 5e^u du$

Explanation:

Part (A)

Step1: Recall integral of constant

The integral of a constant \( k \) with respect to \( x \) is \( kx + C \), where \( C \) is the constant of integration. Here \( k = 7 \).
\( \int 7dx = 7x + C \)

Part (B)

Step1: Use power rule for integration

The power rule for integration is \( \int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (\( n
eq - 1 \)). For \( \int 9x^{2}dx \), we can factor out the constant 9 first.
\( \int 9x^{2}dx=9\int x^{2}dx \)

Step2: Apply power rule

Using the power rule with \( n = 2 \), we have \( \int x^{2}dx=\frac{x^{2 + 1}}{2+1}+C=\frac{x^{3}}{3}+C \). Then multiply by 9.
\( 9\times\frac{x^{3}}{3}+C = 3x^{3}+C \)

Part (C)

Step1: Recall integral of \( \frac{1}{x} \)

The integral of \( \frac{1}{x} \) with respect to \( x \) is \( \ln|x|+C \), where \( C \) is the constant of integration.
\( \int\frac{1}{x}dx=\ln|x| + C \)

Part (D)

Answer:

s:
(A) \( \boldsymbol{7x + C} \)

(B) \( \boldsymbol{3x^{3}+C} \)

(C) \( \boldsymbol{\ln|x| + C} \)

(D) \( \boldsymbol{-\frac{1}{3x^{3}}+C} \)

(E) \( \boldsymbol{5e^{u}+C} \)