# Exercises - Integration (with u-substitution)

1. Integrate

1. $\displaystyle{\int \frac{4}{x^2+6x+9} \ dx}$

2. $\displaystyle{\int \frac{x^2}{(x^3-1)^2} \ dx}$

3. $\displaystyle{\int \frac{x^2}{x^3-1} \ dx}$

4. $\displaystyle{\int \frac{x}{\sqrt{1-4x^4}} \ dx}$

5. $\displaystyle{\int \frac{x^3}{\sqrt{1-4x^4}} \ dx}$

6. $\displaystyle{\int \cos (4x) \ dx}$

7. $\displaystyle{\int e^{3x} \ dx}$

8. $\displaystyle{\int \tan^2 (2x) \ dx}$

9. $\displaystyle{\int \csc \left( \frac{x}{2} \right) \cot \left( \frac{x}{2} \right) \ dx}$

1. $\displaystyle{\int xe^{x^2} \ dx}$

2. $\displaystyle{\int \frac{e^{1/x^2}}{x^3} \ dx}$

3. $\displaystyle{\int \frac{\ln x}{x} \ dx}$

4. $\displaystyle{\int \frac{\ln 3x}{x} \ dx}$

5. $\displaystyle{\int \frac{\ln x^3}{x} \ dx}$

6. $\displaystyle{\int \sin^2(3x) \ dx}$

7. $\displaystyle{\int \cos^3(2x) \ dx}$

8. $\displaystyle{\int x^2 \csc^4 (x^3) \ dx}$

9. $\displaystyle{\int \frac{\sin \sqrt{x}}{\sqrt{x}} \ dx}$

2. Integrate

1. $\displaystyle{\int (x^2-5)^7 \ dx}$

2. $\displaystyle{\int 4\sqrt[3]{1-x} \ dx}$

3. $\displaystyle{\int \frac{4x+6}{(x^2+3x+7)^2} \ dx}$

4. $\displaystyle{\int \cos^3 x \ dx}$

5. $\displaystyle{\int \sin 2x \ dx}$

6. $\displaystyle{\int \sec^2 x \ \tan^2 x \ dx}$

7. $\displaystyle{\int \sin^2 x \ dx}$

8. $\displaystyle{\int (9x^2+6x+1)^{3/2} \ dx}$

9. $\displaystyle{\int \frac{2x}{\sqrt[3]{x^2-4}} \ dx}$

10. $\displaystyle{\int \frac{1}{x^4} \sqrt{\frac{x^3+1}{x^3}} \ dx}$

1. $\displaystyle{\int x^2 \sec^2 x^3 \ dx}$

2. $\displaystyle{\int \frac{\sin \left( \frac{1}{2x} \right)}{x^2} \ dx}$

3. $\displaystyle{\int \frac{(1+\sin x)^4}{\sec x + \tan x} \ dx}$

4. $\displaystyle{\int \frac{\sin x}{1- \sin^2 x} \ dx}$

5. $\displaystyle{\int \frac{2x-1}{(2x^2-2x+1)^2} \ dx}$

6. $\displaystyle{\int \tan^2 (2x) \ dx}$

7. $\displaystyle{\int \frac{ \cos \sqrt{x}}{\sqrt{x}} \ dx}$

8. $\displaystyle{\int \frac{4x}{\sqrt{x^2-4}} \ dx}$

3. Integrate

1. $\displaystyle{\int \frac{(x^2-2)^2}{\sqrt{x}} \ dx}$

2. $\displaystyle{\int \frac{\cos(\frac{1}{x})}{x^2} \ dx}$

3. $\displaystyle{\int \left[\frac{3}{x^2} - \frac{3}{\sqrt{x}} \right] \ dx}$

4. $\displaystyle{\int \tan^3 x \ \sec^2 x \ dx}$

5. $\displaystyle{\int x \sqrt[3]{4-x^2} \ dx}$

6. $\displaystyle{\int \frac{\sin^3 x}{1-\cos^2 x} \ dx}$

7. $\displaystyle{\int \tan(3x) \ \sec(3x) \ dx}$

8. $\displaystyle{\int \frac{4x-6}{\sqrt{x^2 - 3x + 4}} \ dx}$

9. $\displaystyle{\int \frac{1}{x^2 + 4x + 4} \ dx}$

10. $\displaystyle{\int \frac{x^3}{\sqrt{4-x^2}} \ dx}$

1. $\displaystyle{\int \cos^5 x \ dx}$

2. $\displaystyle{\int \frac{(4x^2 - 1)^2}{\sqrt{x}} \ dx}$

3. $\displaystyle{\int \cos (2x) \ dx}$

4. $\displaystyle{\int \cot^2 x \ \csc^2 x \ dx}$

5. $\displaystyle{\int \frac{4x+2}{\sqrt[3]{x^2+x+2}} \ dx}$

6. $\displaystyle{\int x^2 \sqrt{x^3-4} \ dx}$

7. $\displaystyle{\int \frac{4x+10}{\sqrt{x^2+5x+6}} \ dx}$

8. $\displaystyle{\int \frac{\cos (2x)}{\sin^2 (2x)} \ dx}$

9. $\displaystyle{\int \sin^5 x \ dx}$

10. $\displaystyle{\int \tan^2 (3x) \ dx}$

4. Evaluate:

1. $\displaystyle{\int \frac{5}{(7x-2)^2} \ dx}$

2. $\displaystyle{\int \frac{x}{\sqrt[3]{1+x^2}} \ dx}$

3. $\displaystyle{\int \frac{1}{\sqrt{x}(1+\sqrt{x})^2} \ dx}$

4. $\displaystyle{\int \frac{x-2}{(x^2-4x+3)^3} \ dx}$

5. $\displaystyle{\int (3x+2)\sqrt{3x^2+4x+9} \ dx}$

6. $\displaystyle{\int \sqrt[4]{1 - \frac{1}{x}} \left( \frac{1}{x^2} \right) \ dx}$

7. $\displaystyle{\int 3t \ \sec^2(2t^2) \ dt}$

8. $\displaystyle{\int [\sec^2(2x) + x \sec(x^2)\tan(x^2)] \ dx }$

9. $\displaystyle{\int \frac{\cos x}{\sqrt{\sin x}} \ dx}$

10. $\displaystyle{\int \frac{\cos x}{(1-\sin x)^2} \ dx}$

1. $\displaystyle{\int \frac{\cos^2 x}{1-\sin^2 x} \ dx}$

2. $\displaystyle{\int \sqrt{\frac{\sin x}{\cos^5 x}} \ dx}$

3. $\displaystyle{\int \frac{\cos \sqrt[3]{x}}{\sqrt[3]{x^2}} \ dx}$

4. $\displaystyle{\int \frac{dx}{\sqrt{-x^2+6x-5}}}$

5. $\displaystyle{\int \frac{x^3}{\sqrt{x^2-15}} \ dx}$

6. $\displaystyle{\int \frac{e^x}{\sqrt{1-e^{2x}}} \ dx}$

7. $\displaystyle{\int \cos^2 (4x) \sin (4x) \ dx}$

8. $\displaystyle{\int \frac{dx}{x^2+6x+9}}$

9. $\displaystyle{\int x^2 \sqrt[3]{x^3-1} \ dx}$

10. $\displaystyle{\int \left[ \frac{3}{\sqrt{x}} + \frac{\sqrt[3]{x}}{2} + \sec x \tan x \right] \ dx}$

5. Evaluate:

1. $\displaystyle{\int \frac{(x+3)^2}{\sqrt{x}}}$

2. $\displaystyle{\int \left[ \frac{x}{2} + \frac{2}{x^3} + \frac{1}{2x^2} \right] \ dx}$

3. $\displaystyle{\int \cos (3x) \ dx}$

4. $\displaystyle{\int \frac{dx}{x^2+10x+25}}$

5. $\displaystyle{\int \frac{2x}{\sqrt{x^2-4}} \ dx}$

6. $\displaystyle{\int \frac{2}{x^2+8x+16} \ dx}$

7. $\displaystyle{\int \sqrt{x-4} \ dx}$

8. $\displaystyle{\int \left[ \frac{x^2}{3} + \frac{3}{x^2} \right] \ dx}$

9. $\displaystyle{\int (x+1)^2 \ dx}$

10. $\displaystyle{\int \sqrt{\frac{\cos x}{\sin^5 x}} \ dx}$

11. $\displaystyle{\int \frac{e^x}{1+e^{2x}} \ dx}$

12. $\displaystyle{\int \frac{e^{2x}}{1+e^{2x}} \ dx}$

1. $\displaystyle{\int \frac{dx}{1+e^{2x}} \ dx}$

2. $\displaystyle{\int \frac{dx}{\sqrt{x} (1+\sqrt{x})}}$

3. $\displaystyle{\int \frac{dx}{1+\sqrt{x}}}$

4. $\displaystyle{\int x\sqrt{2x+1} \ dx}$

5. $\displaystyle{\int \frac{dx}{4+9x^2} \ dx}$

6. $\displaystyle{\int \frac{1}{x^2+2x+5} \ dx}$

7. $\displaystyle{\int \frac{x}{\sqrt{8-x^2-2x}} \ dx}$

8. $\displaystyle{\int \frac{x}{x^2-2x+17} \ dx}$

9. $\displaystyle{\int \frac{e^x}{e^{2x} - 4e^x + 4} \ dx}$

10. $\displaystyle{\int \frac{\sec^2 5x}{\sqrt{\tan 5x}} \ dx}$

11. $\displaystyle{\int \frac{e^{1/z}}{z^2} \ dz}$

6. (Challenge) Evaluate:

1. $\displaystyle{\int \sin^2 x \ \cos^2 x \ dx}$

2. $\displaystyle{\int \frac{1}{1+e^x} \ dx}$

3. $\displaystyle{\int \frac{\sqrt{x}}{1+\sqrt{x}} \ dx}$

4. $\displaystyle{\int \frac{x}{x^4-6x^2+18} \ dx}$

5. $\displaystyle{\int \frac{dx}{\sqrt{-x^2-4x}}}$

6. $\displaystyle{\int \frac{\tan^2 \sqrt{x}}{\sqrt{x}} \ dx}$

7. $\displaystyle{\int x \ \csc^4(2x^2) \ dx}$

1. $\displaystyle{\int \frac{x^3-2x^2-2x+4}{x^2+x-1} \ dx}$

2. $\displaystyle{\int \frac{x}{1-\sqrt{1-x}} \ dx}$

3. $\displaystyle{\int \frac{\sin^4 x}{\cos^2 x} \ dx}$

4. $\displaystyle{\int \frac{x}{x^2-2x+2} \ dx}$

5. $\displaystyle{\int \frac{x^3}{\sqrt{x^2 + 4}} \ dx}$

6. $\displaystyle{\int_0^1 \frac{x \textrm{ Arctan } x^2}{1+x^4} \ dx }$

7. $\displaystyle{\int \frac{\tan^5 \left( \frac{1}{x} \right)}{x^2} \ dx}$