Exercise Set 13

1. Find the following integrals using a suitable substitution.
   1. \(\int (4x-3)^5 \; dx\)
   2. \(\int 4\sin(3x+2)\; dx\)
   3. \(\int_2^5 \frac{1}{1-2x} \; dx\)
   4. \(\int \frac{1}{\sqrt{(5-3x)^3}}\; dx\)
   5. \(\int \frac{4x}{\sqrt{(x^2+3)^3}}\; dx\)
   6. \(\int 2xe^{x^2-5}\; dx\)
   7. \(\int \frac{\sqrt{x}}{\sqrt{x}}\; dx\)
   8. \(\int \cos(\theta)\sin(\theta)\; d\theta\)
   9. \(\int \frac{\cos(x)}{1+\sin(x)}\; dx\)
2. Determine the following integrals by making use of trigonometric or hyperbolic identities.
   1. \(\displaystyle \int_0^{\pi/4}4\sin(3\theta)\cos(\theta)d\theta\)
   2. \(\displaystyle \int \left( \cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right)\right)^2 \, dx\)
   3. \(\displaystyle \int \sin^3(x)\, dx\)
   4. \(\displaystyle \int \frac{ \cos(2x)} { \cos^2(x) \sin^2(x)} \, dx\)
   5. \(\displaystyle \int \cos(x) \sin(2x) \, dx\)
   6. \(\displaystyle \int \tan(2x) \,dx\)
   7. \(\displaystyle \int \sinh^3(x) \cosh^2(x)\)
3. Determine the following integrals using the method of partial fractions.
   1. \(\displaystyle \int \frac{1}{x^2+x-6} \, dx\)
   2. \(\displaystyle \int \frac{-12u-13}{(2u+1)(u-3)} \, du\)
   3. \(\displaystyle \int \frac{4x^2+13x-4}{2x^2+5x-3}\, dx\)
   4. \(\displaystyle \int \frac{x^3}{x^2-1}\, dx\)
   5. \(\displaystyle \int \frac{x^5-2x^2}{x^2-1} \, dx\)
   6. \(\displaystyle \int \frac{z+1}{(z-1)^2} \, dz\)
   7. \(\displaystyle \int \frac{5y^2}{(y^2+1)(2y-1)} \, dy\)
4. Determine the following integrals using the method of integration by parts.
   1. \(\displaystyle \int x \cos(x) \,dx\)
   2. \(\displaystyle \int t^2 e^{t} \, dt\)
   3. \(\displaystyle \int e^{x}\sin(x) \, dx\)
   4. \(\displaystyle \int (x+1)^2 e^{-2x} \, dx\)
   5. $^{-1}(x) , dx $.