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Mathematics
\[\int\limits_0^\pi x \sin^3 x\ dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
\[\int\limits_0^\pi x \log \sin x\ dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
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\[\int\limits_0^\pi \frac{x \sin x}{1 + \sin x} dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
\[\int\limits_0^\pi \frac{x}{1 + \cos \alpha \sin x} dx, 0 < \alpha < \pi\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
\[\int\limits_0^\pi x \cos^2 x\ dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
\[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
\[\int\limits_{- \pi/2}^{\pi/2} \sin^3 x\ dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
\[\int\limits_{- 1}^1 \log\left( \frac{2 - x}{2 + x} \right) dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
\[\int\limits_{- \pi/4}^{\pi/4} \sin^2 x\ dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
\[\int\limits_0^\pi \log\left( 1 - \cos x \right) dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
\[\int\limits_{- \pi/2}^{\pi/2} \log\left( \frac{2 - \sin x}{2 + \sin x} \right) dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
Evaluate the following integral:
\[\int_{- a}^a \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right)d\theta\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
\[\int\limits_0^2 x\sqrt{2 - x} dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
\[\int\limits_0^1 \log\left( \frac{1}{x} - 1 \right) dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
Evaluate the following integral:
\[\int_{- 1}^1 \left| xcos\pi x \right|dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
\[\int_0^1 | x\sin \pi x | dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
If `f` is an integrable function such that f(2a − x) = f(x), then prove that
\[\int\limits_0^{2a} f\left( x \right) dx = 2 \int\limits_0^a f\left( x \right) dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
If f(2a − x) = −f(x), prove that
\[\int\limits_0^{2a} f\left( x \right) dx = 0 .\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
If f is an integrable function, show that
\[\int\limits_{- a}^a f\left( x^2 \right) dx = 2 \int\limits_0^a f\left( x^2 \right) dx\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
If f is an integrable function, show that
\[\int\limits_{- a}^a x f\left( x^2 \right) dx = 0\]
Chapter: [7] Integrals
Concept: undefined >> undefined
Concept: undefined >> undefined
