IB Mathematics Analysis and Approaches Topic SL…

A function $g(x)$ is defined for $x>0$ with derivative $g^{\prime}(x)=\frac{4}{x \sqrt{x}}$ . The region enclosed by the graph of , the x-axis, the line , and the line is rotated about the x-axis. The resulting volume is cubic units.

(a) Find $\int \frac{4}{x \sqrt{x}} d x$ .

(b) Find $g(x)$ , given that $g(1)=0$ .

A function $h(x)$ for $x \geq 0$ has derivative $h^{\prime}(x)=\frac{8 x}{\left(2 x^{2}+1\right)^{2}}$ .

(a) Show that $\int \frac{8 x}{\left(2 x^{2}+1\right)^{2}}\,dx=-\frac{2}{2 x^{2}+1}+c$ .

(b) Given that $h(0)=3$ , find $h(x)$ .

Consider a function $f(x)$ for $x>0$ . The derivative is given by $f^{\prime}(x)=\frac{2 x}{x^{2}+1}$ .

(a) Show that $\int \frac{2 x}{x^{2}+1}\,d x=\ln \left(x^{2}+1\right)+c$ .

(b) Given that $f(1)=2$ , find $f(x)$ .

IB Mathematics Analysis and Approaches SL 5.10—indefinite Integration, Reverse Chain, by Substitution Questionbank

IB Mathematics Analysis and Approaches SL 5.10—indefinite Integration, Reverse Chain, by Substitution Questionbank

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IB Mathematics Analysis and Approaches SL 5.10—indefinite Integration, Reverse Chain, by Substitution Questionbank

IB Mathematics Analysis and Approaches SL 5.10—indefinite Integration, Reverse Chain, by Substitution Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 5.10—indefinite Integration, Reverse Chain, by Substitution Questionbank

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