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Find the following:
(a)
int16x^(-3)dx quad(x!=0)
(d)
int2e^(-2x)dx
(b)
int9x^(8)dx
(e)
int(4x)/(x^(2)+1)dx
(c)
int(x^(5)-3x)dx
(f)
int(2ax+b)(ax^(2)+bx)^(7)dx

$1$ . Find the following: $\newline$ (a) $\int 16 x^{-3} d x \quad(x \neq 0)$ $\newline$ (d) $\int 2 e^{-2 x} d x$ $\newline$ (b) $\int 9 x^{8} d x$ $\newline$ (e) $\int \frac{4 x}{x^{2}+1} d x$ $\newline$ (c) $\int\left(x^{5}-3 x\right) d x$ $\newline$ (f) $\int(2 a x+b)\left(a x^{2}+b x\right)^{7} d x$

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Find derivatives of using multiple formulae

Full solution

Q.

1

. Find the following:

\newline

(a)

\int 16 x^{-3} d x \quad(x \neq 0)

\newline

(d)

\int 2 e^{-2 x} d x

\newline

(b)

\int 9 x^{8} d x

\newline

(e)

\int \frac{4 x}{x^{2}+1} d x

\newline

(c)

\int\left(x^{5}-3 x\right) d x

\newline

(f)

\int(2 a x+b)\left(a x^{2}+b x\right)^{7} d x

Evaluate Integral: (a) Evaluate the integral of $16x^{-3}$ with respect to $x$ . $\newline$ **Reasoning:** Use the power rule for integration, $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$ . $\newline$ **Calculation:** $\int 16x^{-3} \, dx = 16 \cdot \frac{x^{-3+1}}{-3+1} + C = 16 \cdot \frac{x^{-2}}{-2} + C = -8x^{-2} + C$ . $\newline$ **Math Error Check:**
Apply Power Rule: (b) Evaluate the integral of $9x^8$ with respect to $x$ . $\newline$ **Reasoning:** Apply the power rule for integration. $\newline$ **Calculation:** $\int 9x^8 \, dx = 9 \cdot \frac{x^{8+1}}{8+1} + C = 9 \cdot \frac{x^9}{9} + C = x^9 + C$ . $\newline$ **Math Error Check:**
Integrate Each Term: (c) Evaluate the integral of $x^5 - 3x$ with respect to $x$ . $\newline$ **Reasoning:** Integrate each term separately. $\newline$ **Calculation:** $\int (x^5 - 3x) \, dx = \int x^5 \, dx - \int 3x \, dx = \frac{x^6}{6} - 3 \cdot \frac{x^2}{2} + C = \frac{x^6}{6} - \frac{3x^2}{2} + C$ . $\newline$ **Math Error Check:**
Use Integration Rule: (d) Evaluate the integral of $2e^{-2x}$ with respect to $x$ . $\newline$ **Reasoning:** Use the integration rule for $e^{kx}$ , $\int e^{kx} \, dx = \frac{e^{kx}}{k} + C$ . $\newline$ **Calculation:** $\int 2e^{-2x} \, dx = 2 \cdot \frac{e^{-2x}}{-2} + C = -e^{-2x} + C$ . $\newline$ **Math Error Check:**
Recognize Derivative: (e) Evaluate the integral of $\frac{4x}{x^2+1}$ with respect to $x$ . $\newline$ **Reasoning:** Recognize the derivative of the denominator in the numerator. $\newline$ **Calculation:** $\int \frac{4x}{x^2+1} \, dx = 2 \ln(x^2+1) + C$ . $\newline$ **Math Error Check:**
Use Substitution: (f) Evaluate the integral of $(2ax+b)(ax^2+bx)^7$ with respect to $x$ . $\newline$ **Reasoning:** Use substitution, let $u = ax^2 + bx$ . Then, $du = (2ax + b)dx$ . $\newline$ **Calculation:** $\int (2ax+b)(ax^2+bx)^7 \, dx = \int u^7 \, du = \frac{u^8}{8} + C$ . $\newline$ **Math Error Check:**

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