U-Substitution: Evaluate the following integrals.

  1. \(\displaystyle{ \int (1-2x)^5 \, dx }\)  Solution

  2. \(\displaystyle{ \int \frac{x}{x+1} \, dx }\)  Solution

  3. \(\displaystyle{ \int x \, \cos (1+x^2) \, dx }\)  Solution

  4. \(\displaystyle{ \int \frac{x^2}{1+x^3} \, dx }\)  Solution

  5. \(\displaystyle{ \int \frac{(\ln x)^4}{4x} \, dx }\)  Solution

  6. \(\displaystyle{ \int \frac{x^2-4}{x^3-12x+6} \, dx }\)  Solution

  7. \(\displaystyle{ \int \frac{\sin (\ln x)}{x} \, dx }\)  Solution

  8. \(\displaystyle{ \int \cot \theta \, d\theta }\)  Solution

  9. \(\displaystyle{ \int \tan x \, dx }\)  Solution

  10. \(\displaystyle{ \int x^2 \cos (x^3) \, dx }\)  Solution

  11. \(\displaystyle{ \int (t-2)\sqrt{3t^2-12t} \, dt }\)  Solution

  12. \(\displaystyle{ \int \left(1-\frac{1}{x}\right)\, \sin\left(x - \ln x \right) \, dx }\)  Solution

  1. \(\displaystyle{ \int e^{x^5} x^4 \, dx }\)  Solution

  2. \(\displaystyle{ \int \frac{x^8}{\sqrt{x^9-7}} \, dx }\)  Solution

  3. \(\displaystyle{ \int \frac{e^{5x}}{2+e^{5x}} \, dx }\)  Solution

  4. \(\displaystyle{ \int \frac{1}{x \ln x} \, dx }\)  Solution

  5. \(\displaystyle{ \int x \, \sqrt{1-x^2} \, dx }\)  Solution

  6. \(\displaystyle{ \int \sin \theta \, (1+ \cos \theta)^3 \, d\theta }\)  Solution

  7. \(\displaystyle{ \int_{-1}^1 x\,(x^2+1)^2 \, dx }\)  Solution

  8. \(\displaystyle{ \int_0^1 x \, \sqrt{5x^2+4} \, dx }\)  Solution

  9. \(\displaystyle{ \int_1^2 \frac{x}{(x^2+3)^2} \, dx }\)  Solution

  10. \(\displaystyle{ \int_0^1 \frac{x+2}{\sqrt{x^2+4x}} \, dx }\)  Solution

  11. \(\displaystyle{ \int_0^4 \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx }\)  Solution

  12. \(\displaystyle{ \int_0^1 \frac{\tan^{-1} t}{1+t^2} \, dx }\)  Solution

  13. \(\displaystyle{ \int \frac{x^3}{(1+x^2)^2} \, dx }\)  Solution

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