Litery a, b, c będą zastąpione konkretnymi liczbami, do każdej całki nieoznaczonej mogą być dodane granice

Proste przykłady – podstawowe wzory

1. ∫e^4 − xdx

2. $\int_{}^{}\frac{sin3x + 5x^{14}}{cos3x - x^{15} + ln7}$dx

3. $\int_{}^{}{\left( \sqrt{x} + \frac{1}{\sqrt[3]{x}} \right)^{2}\text{dx}}$

4. $\int_{}^{}\frac{ax^{2} + b}{x^{2} + 1}\text{dx}$

5. ∫tgxdx

6. ∫tg²xdx

7. $\int_{}^{}{\sqrt{x\sqrt{x\sqrt{x}}}\text{dx}}$

8. $\int_{}^{}2^{x}\sqrt[5]{8^{x}}\text{dx}$

9. $\int_{}^{}\frac{2\sqrt[3]{x} + 3\sqrt{x^{7}} - 4\sqrt[4]{x^{11}}}{\sqrt[5]{x^{2}}}\text{dx}$

10. $\int_{}^{}\frac{\text{cosx}}{5sinx + 3}\text{dx}$

11. $\int_{}^{}\frac{cosx + 2x^{5} - 3}{\sqrt{sin3x + x^{6} - 9x + 34}}\text{dx}$

12. $\int_{}^{}\frac{e^{5x} - 3cosx + arctgx}{\sqrt{arcsinx + ln3x - 15}}\text{dy}$

Całkowanie przez części

1. $\int_{}^{}\frac{\text{lnx}}{x^{a}}\text{dx}$

2. $\int_{}^{}{\sqrt[a]{x}\text{lnx}}\text{dx}$

3. ∫xarctgxdx

4. ∫lnxdx

5. $\int_{}^{}\frac{\ln^{2}\text{xdx}}{x^{2}}$

6. $\int_{}^{}\frac{\text{tgxdx}}{\text{lncosx}}$

7. ∫(arccosx)²dx

8. ∫e^axcosbxdx

9. ∫xsinaxdx

10. ∫arcctgxdx

11. ∫xln²xdx

12. ∫arcsinxdx

13. $\int_{}^{}{\frac{x}{\cos^{2}x}\text{dx}}$

14. $\int_{}^{}\frac{\text{arcosxdx}}{\sqrt{x + 1}}$

15. ∫xln(1−x)dx

16. ∫ln(1+x²)dx

Całkowanie przez podstawienie

1. $\int_{}^{}\frac{\left( \text{lnx} \right)^{a}}{x}\text{dx}$

2. ∫(7x+8)³⁵dx

3. $\int_{}^{}\frac{\text{dx}}{x\sqrt{1 - lnx}}$

4. $\int_{}^{}\frac{x^{2}\text{dx}}{e^{x^{3}}}$

5. $\int_{}^{}\frac{3^{x}\text{dx}}{1 + 9^{x}}$

6. $\int_{}^{}\frac{x^{2}\text{dx}}{\sqrt{1 - x^{6}}}$

7. ∫cosaxe^sinaxdx

8. ∫xe^x2 − 5dx

9. $\int_{}^{}\frac{\text{dx}}{\sin^{2}x + 4\cos^{2}x}$ podstawienie t = tgx

10. $\int_{}^{}\frac{\text{dx}}{{3sin}^{2}x + 1}$ podstawienie t = tgx

11. $\int_{}^{}\frac{x^{2}}{\sqrt{1 - x^{2}}}\text{dx}$ podstawienie x = sint

12. $\int_{}^{}\frac{\sqrt{\left( {1 - x}^{2} \right)^{3}}}{x^{6}}\text{dx}$ podstawienie x = sint

13. ∫tg³xdx

14. ∫xe^x2 + 5dx

15. $\int_{}^{}\frac{\left( \text{arctgx} \right)^{a}}{1 + x^{2}}$dx

Rozkład na ułamki proste

1. $\int_{}^{}{\frac{3x}{\left( x - 1 \right)^{2}}\text{dx}}$

2. $\int_{}^{}{\frac{3x^{2} + x - 2}{\left( x - 1 \right)^{2}\left( x^{2} + 1 \right)}\text{dx}}$

3. $\int_{}^{}\frac{1}{x^{2} + 2x - 3}$dx

4. $\int_{}^{}\frac{\text{dx}}{x^{2} + 4x + 13}$

5. $\int_{}^{}{\frac{31}{x^{3}(x - 1)}\text{dx}}$

6. $\int_{}^{}\frac{x - 4}{\left( x - 2 \right)\left( x - 3 \right)}$dx

7. $\int_{}^{}\frac{2x + 7}{x^{2} + x - 2}$dx

8. $\int_{}^{}\frac{3x^{2} + 2x - 3}{x^{3} - x}$dx

9. $\int_{}^{}\frac{\text{dx}}{{x - 5x}^{2}}$

10. $\int_{}^{}\frac{x + 2}{x^{3} - 2x^{2}}$dx

11. $\int_{}^{}\frac{x^{2} + 2x + 6}{\left( x - 2 \right)\left( x - 1 \right)\left( x - 3 \right)}$dx

12. $\int_{}^{}\frac{{3x}^{2} + 8}{x^{3} + 4x^{2} + 4x}\text{dx}$

13. $\int_{}^{}\frac{x^{2} + 1}{x^{3} - 3x^{2} + 3x - 1}\text{dx}$

14. $\int_{}^{}\frac{x^{2}}{x^{4} + 5x^{2} + 1}\text{dx}$

15. $\int_{}^{}\frac{(4x - 3)dx}{x^{2} + 8x + 25}$

Funkcje trygonometryczne

1. ∫sin⁴axdx

2. ∫sin³axdx

3. ∫sin⁵axdx

4. ∫cos²axdx

5. ∫sin⁴axcos³xdx

6. ∫sinaxsinbxcoscxdx

7. ∫sinaxsinbxdx

8. ∫cosaxsinbxdx

9. ∫cosaxcosbxdx

10. $\int_{}^{}\frac{\text{dx}}{\text{cosx}}$

11. $\int_{}^{}\frac{\text{dx}}{a + cosx}$ Podstawienie elementarne t = tg(x/2)

12. $\int_{}^{}\frac{\text{dx}}{\text{asin}x + bcosx}$ Podstawienie elementarne t = tg(x/2)

Pierwiastki

1. $\int_{}^{}\sqrt{1 - a^{2}x^{2}}$dx

2. $\int_{}^{}\frac{1}{x^{2}\sqrt{4 - x^{2}}}\text{dx}$

3. $\int_{}^{}\frac{x^{3}}{\sqrt{3 - 4x + x^{2}}}\text{dx}$

4. $\int_{}^{}\frac{1}{x - \sqrt{x}}\text{dx}$

5. $\int_{}^{}\frac{1}{\sqrt{x^{2} + 10x - 15}}\text{dx}$

6. $\int_{}^{}\frac{1}{\sqrt{x^{2} + 6x + 15}}\text{dx}$

7. $\int_{}^{}\frac{5x - 3}{\sqrt{{2x}^{2} + 8x + 1}}\text{dx}$

8. $\int_{}^{}\frac{\text{dx}}{\sqrt{1 - 2x} - \sqrt[4]{1 - 2x}}$

9. $\int_{}^{}\frac{1}{x}\sqrt{\frac{x + 2}{x}}\text{dx}$

10. $\int_{}^{}{\sqrt{\frac{x + 1}{x - 1}}\text{dx}}$

11. $\int_{}^{}\frac{1}{x^{2}}\sqrt{\frac{x + 1}{x}}\text{dx}$

12. $\int_{}^{}{x\sqrt{x + a}\text{dx}}$

13. $\int_{}^{}{\frac{1 + \sqrt[4]{x}}{x + \sqrt{x}}\text{dx}}$

14. $\int_{}^{}\frac{\text{dx}}{\left( 1 + \sqrt[3]{x} \right)\sqrt{x}}$

15. $\int_{}^{}\sqrt{1 + a^{2}x^{2}}$dx

Całki oznaczone

1. $\int_{0}^{6}{\frac{x}{\sqrt{4 + x^{2}}}\text{dx}}$

2. $\int_{0}^{2}\frac{e^{\text{ax}}}{1 + e^{\text{bx}}}\text{dx}$

3. ∫₀¹xe^−xdx

4. ∫₀¹x

5. $\int_{ln2}^{ln2}\frac{e^{x} - 1}{e^{x} + 1}\ $= 0 Dlaczego ?

6. $\int_{- 1}^{1}\frac{2x^{5} - x^{3} + x}{x^{2} + 1}dx = 0$ Dlaczego ?

Całki niewłaściwe

1. $\int_{- \infty}^{+ \infty}{\frac{1}{a^{2} + x^{2}}\text{dx}}$

2. $\int_{0}^{\frac{\pi}{2}}\text{ctgxdx}$

3. $\int_{0}^{\frac{\pi}{2}}\text{tgaxdx}$

4. $\int_{- 1}^{2}\frac{\text{dx}}{\sqrt[3]{\left( x - 1 \right)^{2}}}$

5. $\int_{0}^{2}\frac{x^{3}\text{dx}}{\sqrt{4 - x^{2}}}$ podstawienie x = 2sint

6. $\int_{1}^{\infty}{\frac{\text{lnx}}{x^{3}}\text{dx}}$

7. ∫_−∞⁰xe^axdx

8. $\int_{- 2}^{2}\frac{\text{xdx}}{x^{2} - 1}$

Zadania różne

1. ∫sinlnx²dx

2. ∫sinlnxdx

3. $\int_{}^{}{\text{arctg}\sqrt{x}\text{dx}}$

4. $\int_{}^{}{e^{\sqrt{x}}\text{dx}}$

5. $\int_{}^{}{e^{\sqrt[3]{x}}\text{dx}}$

6. $\int_{}^{}{\frac{1}{1 + tgx}\text{dx}}$

7. $\int_{}^{}\frac{\text{dx}}{\sqrt{x + 5} + \sqrt{x}}$

Pole obszaru ograniczonego krzywymi

1. y = x² – b i y = ax

3. y = ax - x² i y = bx

8. y = x³ , y = ax

9. y = - x²       x =  y²

10. y = x² y = 2x² y =8