Definite Integrals

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3 Problem: Calculate the integral: \[ \int_{-2 \pi}^{2 \pi} \sin ^{3} 2 x \sqrt{100-x^{2}} d x \]

9.3.3.1 Definite Integrals

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(a) Problem: Calculate the integral: \[ \int_{0}^{1} \frac{\sqrt{x} d x}{4-x} \]

9.3.3.2 Definite Integrals

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Problem: Calculate the integral: 1) \( \int_{-3}^{3} \frac{x^{2} \sin 2 x}{x^{2}+1} d x \) 2) \( \int_{0}^{2} \frac{x^{5} d x}{\sqrt{4-x^{2}}} \).

9.3.3.3 Definite Integrals

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3) Problem: Calculate the definite integral: \[ \int_{5 / 2}^{10 / 3} \frac{\sqrt{x+2}+\sqrt{x-2}}{(\sqrt{x+2}-\sqrt{x-2})(x-2)^{2}} d x . \]

9.3.3.4 Definite Integrals

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T Problem: Calculate the definite integral: \[ \int_{0}^{\sqrt{3}} \frac{d x}{\sqrt{x\left(4-x^{2}\right)^{3}}} \]

9.3.3.5 Definite Integrals

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3) Problem: Calculate the definite integral: \[ \int_{1}^{e^{\pi / 2}} \cos \ln x d x \]

9.3.3.6 Definite Integrals

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30 Problem: Calculate the definite integral: \[ \int_{0}^{1} \frac{4 x^{3}+11 x^{2}+12 x+4}{(x+1)^{2}\left(x^{2}+2 x+2\right)} d x . \]

9.3.3.7 Definite Integrals

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3) Problem: Write down the integral sum for the function \( f(x)=3 x+1 \) on the segment \( [a, b]=[-1,2] \).

9.3.3.8 Definite Integrals

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(3) Problem: Using the definite integral, calculate the limit of the sequence \[ S_{n}=\frac{1^{4}+2^{4}+\cdots+n^{4}}{n^{5}} \quad \text { when } n \rightarrow \infty \]

9.3.3.9 Definite Integrals

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Problem: Prove the equality: \[ \int_{0}^{1} x^{m}(1-x)^{n} d x=\int_{0}^{1}(1-x)^{m} x^{n} d x . \]

9.3.3.10 Definite Integrals

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3) Problem: Calculate the definite integral: \[ \int_{5 / 2}^{10 / 3} \frac{\sqrt{x+2}+\sqrt{x-2}}{(\sqrt{x+2}-\sqrt{x-2})(x-2)^{2}} d x \]

9.3.3.11 Definite Integrals

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(3) Problem: Calculate the definite integral: \[ \int_{0}^{\sqrt{3}} \frac{d x}{\sqrt{\left(4-x^{2}\right)^{3}}} \]

9.3.3.12 Definite Integrals

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3) Problem: Find the value of the definite integral: \[ \int_{\ln 3}^{\ln 8} \frac{d x}{\sqrt{e^{x}+1}} \]

9.3.3.13 Definite Integrals

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3o Problem: Calculate the definite integral: \[ \int_{0}^{1} \frac{\sin ^{-1} x}{\sqrt{1+x}} d x . \]

9.3.3.14 Definite Integrals

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(2) Problem: Find the value of the definite integral: \[ \int_{0}^{\sqrt{5}} \sqrt{15-3 \mathrm{x}^{2}} d x \]

9.3.3.15 Definite Integrals

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