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Home Question Bank Math Calculus Archive 2024 December 30

Calculus Questions from Dec 30,2024

Browse the Calculus Q&A Archive for Dec 30,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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xercice 1 Soit A un nombre réel tel que : \[ A=\ln (2+\sqrt{2})+\ln (2+\sqrt{2+\sqrt{2}})+\ln (2-\sqrt{2+\sqrt{2}}) \] Montrer que: \( A=\ln 2 \) Calculer: \( \lim _{x \rightarrow+\infty} \frac{\ln (3 x+1)}{\ln x} \) a) Résoudre dans \( \mathbb{R} \) Céquation: (E): \( x^{2}-3 x+2=0 \) b) déduire les sofutions de Céquation \( \left(E_{1}\right): \ln ^{2}(x)-3 \ln (x)+2=0 \) c) déduire les solutions de Cinéquation \( (I): \ln ^{2}(x)-3 \ln (x)>-2 \) a) montrer que: \[ \forall x, y \in] 0 ;+\infty\left[, \frac{x+y}{2} \geq \sqrt{x y}\right. \] b) déduire que : \[ \forall x, y \in] 0 ;+\infty\left[, \quad \ln \left(\frac{x+y}{2}\right) \geq \frac{\ln x+\ln y}{2}\right. \] xercice \( 2^{\circ} \) Evaluate the following integral: \( \int x \sin \left(x^{2}+4\right) d x \) \( \ddot { y } - 2 y ^ { \prime } + y = x e ^ { x } \sin x \) \( f(x)=\int_{4}^{x} \frac{5}{t}+\cos (t) d t \) What is \( \frac{d f}{d x} ? \) Suppose that \( f(x)=\int_{2}^{x} \cos (t-2) d t \) What is \( f^{\prime}(x) ? \) Evaluate the following integrals by method of partial fractions \( \begin{array}{lll}\text { (a) } \int \frac{5 x+2}{3 x^{2}+x-4} d x & \text { (b) } \int \frac{x+1}{4 x^{2}-1} d x & \text { (c) } \int \frac{3 x}{1+x-2 x^{2}} d x\end{array} \) CHAPTER NINE: INTEGRAL CALCULUS 1. Evaluate the integrals \( \begin{array}{lll}\text { (a) } \int x^{2} \cos 2 x d x & \text { (b) } \int x^{4} e^{3 x} d x & \text { (c) } \int x^{4} \log (x+4) d x \\ \text { (d) } \int \frac{\sin ^{-1} x}{\sqrt{1-x^{2}}} d x & \text { (e) } \int e^{2 x} \cos 4 x d x & \text { (f) } \int \sinh 7 x d x \\ \text { (g) } \int \sqrt{7 x-1} d x & \text { (h) } \int x(5 x+6)^{6} d x & \text { (i) } \int x^{2} \sqrt{x+1} d x \\ \begin{array}{lll}\text { (j) } \int x\left(5+x^{2}\right)^{8} d x & \text { (k) } \int x^{2} \sqrt{7-4 x^{3}} d x & \text { (k) } \int \cot x d x \\ \text { (l) } \int \frac{8 x^{3}+9 x^{2}-10 x+6}{2 x^{4}+3 x^{3}-5 x^{2}+6 x+1} d x & \text { (m) } \int \frac{x^{2}}{x^{3}+1} d x\end{array}\end{array} \). 1. Evaluate the integrals \( \begin{array}{lll}\text { (a) } \int x^{2} \cos 2 x d x & \text { (b) } \int x^{4} e^{3 x} d x & \text { (c) } \int x^{4} \log (x+4) d x \\ \text { (d) } \int \frac{\sin ^{-1} x}{\sqrt{1-x^{2}}} d x & \text { (e) } \int e^{2 x} \cos 4 x d x & \text { (f) } \int \sinh 7 x d x \\ \text { (g) } \int \sqrt{7 x-1} d x & \text { (h) } \int x(5 x+6)^{6} d x & \text { (i) } \int x^{2} \sqrt{x+1} d x \\ \text { (j) } \int x\left(5+x^{2}\right)^{8} d x & \text { (k) } \int x^{2} \sqrt{7-4 x^{3}} d x & \text { (k) } \int \cot x d x\end{array} \) \( \begin{array}{lll}\text { (l) } \int \frac{8 x^{3}+9 x^{2}-10 x+6}{2 x^{4}+3 x^{3}-5 x^{2}+6 x+1} d x & \text { (m) } \int \frac{x^{2}}{x^{3}+1} d x\end{array} \) Consider rectangles whose height is given by the value of \( f(x)=\frac{1}{x} \). Using the right-hand method and four intervals, what is the estimated area under the graph between \( x=1 \) and \( x=5 \) ? Round to the nearest hundredth, and use four intenvals. 1.24 Evaluate the integrals \[ \begin{array}{ll}\text { (a) } \int e^{-3 x} \cos 2 x d x & \text { (b) } \int \sin 5 x \cos 3 x d x\end{array} \text { (c) } \int \frac{2 x^{2}+x+1}{(x-1)\left(x^{2}+1\right)} d x \]
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