91Ó°ÊÓ

Find an expression for the integral which contains \(g\) but no integral sign. $$\int g^{\prime}(x) \sin g(x) d x$$

Find an expression for the integral which contains \(g\) but no integral sign. $$\int g^{\prime}(x) \sqrt{1+g(x)} d x$$

Find a substitution \(w\) and constants \(k, n\) so that the integral has the form \(\int k w^{n} d w\). $$\int x^{2} \sqrt{1-4 x^{3}} d x$$

Find a substitution \(w\) and constants \(k, n\) so that the integral has the form \(\int k w^{n} d w\). $$\int \frac{\cos t}{\sin t} d t$$

Find a substitution \(w\) and constants \(k, n\) so that the integral has the form \(\int k w^{n} d w\). $$\int \frac{2 x d x}{\left(x^{2}-3\right)^{2}}$$

Find constants \(k, n, w_{0}, w_{1}\) so the the integral has the form \(\int_{w_{0}}^{w_{1}} k w^{n} d w\). $$\int_{1}^{5} \frac{3 x d x}{\sqrt{5 x^{2}+7}}, \quad w=5 x^{2}+7$$

Find constants \(k, n, w_{0}, w_{1}\) so the the integral has the form \(\int_{w_{0}}^{w_{1}} k w^{n} d w\). $$\int_{0}^{5} \frac{2^{x} d x}{2^{x}+3}, \quad w=2^{x}+3$$

Find a substitution \(w\) and a constant \(k\) so that the integral has the form \(\int k e^{w} d w\). $$\int x e^{-x^{2}} d x$$

Find a substitution \(w\) and a constant \(k\) so that the integral has the form \(\int k e^{w} d w\). $$\int e^{\sin \phi} \cos \phi d \phi$$

Find a substitution \(w\) and a constant \(k\) so that the integral has the form \(\int k e^{w} d w\). $$\int \sqrt{e^{r}} d r$$