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Calculus

Integration Techniques

Concept Quizzes

Integration U-substitution - Given U
Integration U-substitution - Trigonometric
Integration U-substitution - Problem Solving
Integration by Parts
Integration with Partial Fractions

Challenge Quizzes

Integration Techniques: Level 2 Challenges
Integration Techniques: Level 3 Challenges
Integration Techniques: Level 4 Challenges
Integration Techniques: Level 5 Challenges

Integration U-substitution - Problem Solving

When $a=2$ and $b=4,$ $\int_{a}^{b}{\sqrt{(x-a)(b-x)}}dx$ is equal to $\underline{\qquad}.$

\frac{\pi}{2}

\pi

\frac{\pi}{4}

\frac{\pi}{6}

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by Brilliant Staff

For $n$ a positive integer, what is the indefinite integral $\int{ x{(1-x)}^{n+6} }dx?$

Details and assumptions

Use $C$ as the constant of integration.

-\frac{{(1-x)}^{n+7}}{n+7} + \frac{{(1-x)}^{n+8}}{n+8} + C

-\frac{{(1-x)}^{n+7}}{n+8} + \frac{{(1-x)}^{n+9}}{n+10} + C

-\frac{{(1-x)}^{n+6}}{n+8} + \frac{{(1-x)}^{n+9}}{n+10} + C

-\frac{{(1-x)}^{n+7}}{n+8} + \frac{{(1-x)}^{n+9}}{n+7} + C

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by Brilliant Staff

Evaluate $\int_{0}^{\sqrt{3}} (x+4)^2 e^{x^2} dx+\int_{\sqrt{3}}^{0} (x-4)^2 e^{x^2} dx.$

\sqrt{3}(e^{4}-1)

8e^{3}

\sqrt{3}(e^{8}-1)

8(e^{3}-1)

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by Brilliant Staff

Given that $\displaystyle \int_0^4 x^3\sqrt{9+x^2} dx = a$ , what is the value of $\lfloor a \rfloor$ ?

Details and assumptions

Greatest Integer Function: $\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}$ refers to the greatest integer less than or equal to $x$ . For example $\lfloor 2.3 \rfloor = 2$ and $\lfloor -3.4 \rfloor = -4$ .

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by Brilliant Staff

Determine the value of $100 \int_0^{\pi} \sqrt{ 1 + \sin x} \, dx$ .

This problem is proposed by Cody.

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by Brilliant Staff