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Find:
(a)
int13e^(x)dx
(d)
int3e^(-(2x+7))dx
(b)
int(3e^(x)+(4)/(x))dx quad(x > 0)
(e)
int4xe^(x^(2)+3)dx
(c)
int(5e^(x)+(3)/(x^(2)))dx quad(x!=0)
(f)
int xe^(x^(2)19)dx

$2$ . Find: $\newline$ (a) $\int 13 e^{x} d x$ $\newline$ (d) $\int 3 e^{-(2 x+7)} d x$ $\newline$ (b) $\int\left(3 e^{x}+\frac{4}{x}\right) d x \quad(x>0)$ $\newline$ (e) $\int 4 x e^{x^{2}+3} d x$ $\newline$ (c) $\int\left(5 e^{x}+\frac{3}{x^{2}}\right) d x \quad(x \neq 0)$ $\newline$ (f) $\int x e^{x^{2} 19} d x$

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Find derivatives of using multiple formulae

Full solution

Q.

2

. Find:

\newline

(a)

\int 13 e^{x} d x

\newline

(d)

\int 3 e^{-(2 x+7)} d x

\newline

(b)

\int\left(3 e^{x}+\frac{4}{x}\right) d x \quad(x>0)

\newline

(e)

\int 4 x e^{x^{2}+3} d x

\newline

(c)

\int\left(5 e^{x}+\frac{3}{x^{2}}\right) d x \quad(x \neq 0)

\newline

(f)

\int x e^{x^{2} 19} d x

Evaluate Integral of e^x: $\newline$ (a) Evaluate the integral of $e^x$ from $1$ to $3$ . $\newline$ $\int_1^3 e^x \, dx = [e^x]_1^3 = e^3 - e^1$
Find Integral of $3$ e^{-( $2$ x+ $7$ )}: $\newline$ (d) Find the integral of $3e^{-(2x+7)}$ with respect to $x$ . $\newline$ Let $u = -2x - 7$ , then $du = -2dx$ or $dx = -\frac{1}{2}du$ . $\newline$ $\int 3e^{-(2x+7)} \, dx = \int 3e^u \left(-\frac{1}{2}\right) \, du = -\frac{3}{2} \int e^u \, du = -\frac{3}{2}e^u + C = -\frac{3}{2}e^{-(2x+7)} + C$
Solve Integral of $3$ e^x + $4$ /x: $\newline$ (b) Solve the integral of $3e^x + \frac{4}{x}$ for $x > 0$ . $\newline$ $\int (3e^x + \frac{4}{x}) \, dx = 3\int e^x \, dx + 4\int \frac{1}{x} \, dx = 3e^x + 4\ln|x| + C$
Compute Integral of $4$ xe^{x^ $2$ + $3$ }: $\newline$ (e) Compute the integral of $4xe^{x^2+3}$ . $\newline$ Let $u = x^2 + 3$ , then $du = 2x \, dx$ or $dx = \frac{1}{2x}du$ . $\newline$ $\int 4xe^{x^2+3} \, dx = \int 4xe^u \frac{1}{2x} \, du = 2\int e^u \, du = 2e^u + C = 2e^{x^2+3} + C$
Evaluate Integral of $5$ e^x + $3$ /x^ $2$ : $\newline$ (c) Evaluate the integral of $5e^x + \frac{3}{x^2}$ for $x \neq 0$ . $\newline$ $\int (5e^x + \frac{3}{x^2}) \, dx = 5\int e^x \, dx + 3\int x^{-2} \, dx = 5e^x - \frac{3}{x} + C$
Find Integral of xe^{x^ $2$ $19$ }: $\newline$ (f) Find the integral of $xe^{x^2 19}$ . $\newline$ Let $u = x^2 19$ , then $du = 2x \cdot 19 \, dx$ or $dx = \frac{1}{38x}du$ . $\newline$ $\int xe^{x^2 19} \, dx = \int xe^u \frac{1}{38x} \, du = \frac{1}{38}\int e^u \, du = \frac{1}{38}e^u + C = \frac{1}{38}e^{x^2 19} + C$

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