$7)\displaystyle\int (1+x)\cos x\, dx\\ u=x+1 \Rightarrow du=dx\\ dv=\cos x dx \Rightarrow v=\sin x\\ \displaystyle\int (1+x)\cos x\, dx\\=(x+1)\sin x-\displaystyle\int \sin x \, dx\\=(x+1)\sin x+\cos x+C\\ 8)\displaystyle\int\limits^1_0 (x-1)e^{3x} \, dx\\ u=x-1 \Rightarrow du=dx\\ dv=e^{3x} dx \Rightarrow v=\dfrac{1}{3}e^{3x} \\ \displaystyle\int\limits^1_0 (x-1)e^{3x} \, dx\\ =\dfrac{1}{3}(x-1)e^{3x} \Big|^1_0 -\dfrac{1}{3}\displaystyle\int\limits^1_0e^{3x} \, dx\\ =\dfrac{1}{3}-\dfrac{1}{9}\displaystyle\int\limits^1_0e^{3x} \, d(3x)\\ =\dfrac{1}{3}-\dfrac{1}{9}e^{3x}|^1_0\\ =-\dfrac{1}{9}e^{3}+\dfrac{4}{9}\\ 9)\displaystyle\int\dfrac{3x^2-x+5}{x-4} \, dx\\ =\displaystyle\int\dfrac{3x^2-12x+11x-44+49}{x-4} \, dx\\ =\displaystyle\int\left(3x+11+\dfrac{49}{x-4} \right)\, dx\\ =\dfrac{3}{2}x^2+11x+49\ln(|x-4|)+C\\ 10)\displaystyle\int\limits^2_0 \left[f(x)-2g(x)+1 \right]\, dx\\ =\displaystyle\int\limits^2_0f(x) \, dx-2\displaystyle\int\limits^2_0g(x) \, dx+\displaystyle\int\limits^2_0 \, dx\\ =8-2(-3)+x\Big|^2_0\\ =16$
