Integration by parts and partial fractions

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Find $F$ with $F'(x) = 8x e^x$ and $F(0) = 0$ .
Evaluate $\displaystyle\int_0^{\pi/2} 8x\cos x\,dx$ .
Evaluate $\displaystyle\int_1^{e^{ 3 } } \ln x\,dx$ .
Find $F$ with $F'(x) = 5x^2 e^x$ and $F(0) = 0$ .
To integrate $\displaystyle\int x^{ 1 } \ln x\,dx$ by parts, which choice works?
1. $u = \ln x$ , $dv = x^{ 1 }\,dx$
2. $u = x^{ 1 }$ , $dv = \ln x\,dx$
3. $u = x^{ 1 }\ln x$ , $dv = dx$
To integrate $\displaystyle\int x^{ 2 } \ln x\,dx$ by parts, which choice works?
1. $u = \ln x$ , $dv = x^{ 2 }\,dx$
2. $u = x^{ 2 }$ , $dv = \ln x\,dx$
3. $u = x^{ 2 }\ln x$ , $dv = dx$
To integrate $\displaystyle\int x^{ 4 } \ln x\,dx$ by parts, which choice works?
1. $u = \ln x$ , $dv = x^{ 4 }\,dx$
2. $u = x^{ 4 }$ , $dv = \ln x\,dx$
3. $u = x^{ 4 }\ln x$ , $dv = dx$
To integrate $\displaystyle\int x^{ 5 } \ln x\,dx$ by parts, which choice works?
1. $u = \ln x$ , $dv = x^{ 5 }\,dx$
2. $u = x^{ 5 }$ , $dv = \ln x\,dx$
3. $u = x^{ 5 }\ln x$ , $dv = dx$
To integrate $\displaystyle\int x^{ 3 } \ln x\,dx$ by parts, which choice works?
1. $u = \ln x$ , $dv = x^{ 3 }\,dx$
2. $u = x^{ 3 }$ , $dv = \ln x\,dx$
3. $u = x^{ 3 }\ln x$ , $dv = dx$
Evaluate $\displaystyle\int_0^1 x e^{ 2x}\,dx$ .
$\dfrac{ 8 }{(x - 4)(x - 6)} = \dfrac{A}{x - 4} + \dfrac{B}{x - 6}$ . Find $A$ .
Evaluate $\displaystyle\int_1^2 \dfrac{dx}{x(x + 3)}$ .
Evaluate $\displaystyle\int_{ 5 }^{ 6 } \dfrac{dx}{x^2 - 1}$ .
Which method suits $\displaystyle\int x e^{ 2x^2}\,dx$ best?
1. Substitution, $u = 2x^2$
2. Integration by parts
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 5x^2}\,dx$ best?
1. Substitution, $u = 5x^2$
2. Integration by parts
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 9x^2}\,dx$ best?
1. Substitution, $u = 9x^2$
2. Integration by parts
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 3x^2}\,dx$ best?
1. Substitution, $u = 3x^2$
2. Integration by parts
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 8x^2}\,dx$ best?
1. Substitution, $u = 8x^2$
2. Integration by parts
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 7x^2}\,dx$ best?
1. Substitution, $u = 7x^2$
2. Integration by parts
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 6x^2}\,dx$ best?
1. Substitution, $u = 6x^2$
2. Integration by parts
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 4x^2}\,dx$ best?
1. Substitution, $u = 4x^2$
2. Integration by parts
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 6x}\,dx$ best?
1. Integration by parts, $u = x$
2. Substitution, $u = 6x$
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 3x}\,dx$ best?
1. Integration by parts, $u = x$
2. Substitution, $u = 3x$
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 4x}\,dx$ best?
1. Integration by parts, $u = x$
2. Substitution, $u = 4x$
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 8x}\,dx$ best?
1. Integration by parts, $u = x$
2. Substitution, $u = 8x$
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 5x}\,dx$ best?
1. Integration by parts, $u = x$
2. Substitution, $u = 5x$
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 9x}\,dx$ best?
1. Integration by parts, $u = x$
2. Substitution, $u = 9x$
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 7x}\,dx$ best?
1. Integration by parts, $u = x$
2. Substitution, $u = 7x$
3. Partial fractions
Which method suits $\displaystyle\int x e^{ 2x}\,dx$ best?
1. Integration by parts, $u = x$
2. Substitution, $u = 2x$
3. Partial fractions
Which method suits $\displaystyle\int \dfrac{dx}{x^2 - 36}$ best?
1. Partial fractions: $x^2 - 36 = (x - 6)(x + 6)$
2. The arctangent formula
3. Integration by parts
Which method suits $\displaystyle\int \dfrac{dx}{x^2 - 81}$ best?
1. Partial fractions: $x^2 - 81 = (x - 9)(x + 9)$
2. The arctangent formula
3. Integration by parts
Which method suits $\displaystyle\int \dfrac{dx}{x^2 - 4}$ best?
1. Partial fractions: $x^2 - 4 = (x - 2)(x + 2)$
2. The arctangent formula
3. Integration by parts
Which method suits $\displaystyle\int \dfrac{dx}{x^2 - 64}$ best?
1. Partial fractions: $x^2 - 64 = (x - 8)(x + 8)$
2. The arctangent formula
3. Integration by parts
Which method suits $\displaystyle\int \dfrac{dx}{x^2 - 49}$ best?
1. Partial fractions: $x^2 - 49 = (x - 7)(x + 7)$
2. The arctangent formula
3. Integration by parts
Which method suits $\displaystyle\int \dfrac{dx}{x^2 - 25}$ best?
1. Partial fractions: $x^2 - 25 = (x - 5)(x + 5)$
2. The arctangent formula
3. Integration by parts
Which method suits $\displaystyle\int \dfrac{dx}{x^2 - 16}$ best?
1. Partial fractions: $x^2 - 16 = (x - 4)(x + 4)$
2. The arctangent formula
3. Integration by parts
Which method suits $\displaystyle\int \dfrac{dx}{x^2 - 9}$ best?
1. Partial fractions: $x^2 - 9 = (x - 3)(x + 3)$
2. The arctangent formula
3. Integration by parts

Answers

$8 \left(x - 1\right) e^{x} + 8$
$-8 + 4 \pi$
$1 + 2 e^{3}$
$5 \left(x^{2} - 2 x + 2\right) e^{x} - 10$
$u = \ln x$ , $dv = x^{ 1 }\,dx$
$\frac{1}{4} + \frac{e^{2}}{4}$
$-4$
$\frac{\ln{\left(\frac{8}{5} \right)}}{3}$
$\frac{\ln{\left(\frac{15}{14} \right)}}{2}$
Substitution, $u = 2x^2$
Integration by parts, $u = x$
Partial fractions: $x^2 - 36 = (x - 6)(x + 6)$