91Ó°ÊÓ

Find the integrals .Check your answers by differentiation. $$\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x$$

Decide which function is an antiderivative of the other. $$f(x)=1-\frac{1}{x^{2}} ; g(x)=\frac{1}{x}+x$$

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=\) \(f(x)\) and \(F(0)=0 .\) Is there only one possible solution? $$f(x)=3$$

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=\) \(f(x)\) and \(F(0)=0 .\) Is there only one possible solution? $$f(x)=2+4 x+5 x^{2}$$

Find the integrals .Check your answers by differentiation. $$\int \frac{e^{t}}{e^{t}+1} d t$$

Find the integrals .Check your answers by differentiation. $$\int \sin ^{6}(5 \theta) \cos (5 \theta) d \theta$$

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=\) \(f(x)\) and \(F(0)=0 .\) Is there only one possible solution? $$f(x)=\frac{1}{4} x$$

Find an antiderivative \(F(x)\) with \(F^{\prime}(x)=\) \(f(x)\) and \(F(0)=0 .\) Is there only one possible solution? $$f(x)=\sqrt{x}$$

Find the integrals .Check your answers by differentiation. $$\int \frac{x \cos \left(x^{2}\right)}{\sqrt{\sin \left(x^{2}\right)}} d x$$

If appropriate, evaluate the following integrals by substitution. If substitution is not appropriate, say so, and do not evaluate. (a) \(\int x \sin \left(x^{2}\right) d x\) (b) \(\int x^{2} \sin x d x\) (c) \(\int \frac{x^{2}}{1+x^{2}} d x\) (d) \(\int \frac{x}{\left(1+x^{2}\right)^{2}} d x\) (e) \(\int x^{3} e^{x^{2}} d x\) (f) \(\int \frac{\sin x}{2+\cos x} d x\)