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Integration

1
Introduction to Integration

Problem  1

Integrate.

  1. \(\displaystyle\int{x^3} \,\text{d}x\)

  2. \(\displaystyle\int_{0}^{3}{x^3} \,\text{d}x\)

  3. \(\displaystyle\int_{0}^{4}{x^3} \,\text{d}x\)

  4. \(\displaystyle\int_{0}^{3}{\sqrt{x}} \,\text{d}x\)

  5. \(\displaystyle\int_{\pi/4}^{\pi/4}{\sin{t}} \,\text{d}t\)

  6. \(\displaystyle\int_{0}^{4}{e^x} \,\text{d}x\)

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Problem  2

An example of an AP Calculus Exam question: Multipole Choice

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2
Integrals that Result in Inverse Trigonometric Functions

Problem  1

Integrate.

  1. \(\displaystyle\int\frac{1}{1+x^2}\text{d}x\)

  2. \(\displaystyle\int_{0}^{0.5}\frac{\text{d}x}{\sqrt{1-x^2}}\)

  3. \(\displaystyle\int_{0}^{\sqrt{3}}\frac{6}{1+x^2}\text{d}x\)

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3
Integration by Substitution

Problem  1

Integrate.

  1. \(\displaystyle\int{3x^2\sqrt{1+x^3}\text{d}x}\)

  2. \(\displaystyle\int{\frac{(\ln x)^2}{x}\text{d}x}\)

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Problem  2

Integrate.

  1. \(\displaystyle\int{xe^{x^2}}\text{d}x\)

  2. \(\displaystyle\int{\frac{\text{d}x}{5-3x}}\)

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Problem  3

Integrate.

  1. \(\displaystyle\int{3t^2\sin{t^3}\text{d}x}\)

  2. \(\displaystyle\int{x^4\sqrt{x^5+3}\text{d}x}\)

  3. \(\displaystyle\int{x^3\sqrt{x^5+3}\text{d}x}\)

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Problem  4

Integrate.

  1. \(\displaystyle\int\frac{\text{d}x}{1+x^2}\)

  2. \(\displaystyle\int\frac{x\text{d}x}{1+x^2}\)

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

  4. \(\displaystyle\int\frac{x^2\text{d}x}{1+x^2}\)

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4
Integration by Parts

Problem  1

Integrate \(\displaystyle\int{xe^x\text{d}x}\)

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Problem  2

Integrate \(\displaystyle\int{x\sin{x}\,\text{d}x}\)

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Problem  3

Integrate \(\displaystyle\int{x^3\ln x}\, \text{d}x\)

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Problem  4

Integrate \(\displaystyle\int_0^1{te^{-t}}\, \text{d}t\)

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5
Trigonometric Substitutions

Problem  1

Integrate.

  1. \(\displaystyle\int\displaystyle\frac{1}{\sqrt{1-x^2}}\text{d}x\)

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

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Problem  2

Integrate \(\displaystyle\int\displaystyle\frac{1}{x^2+9}\text{d}x\)

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