Chapter 6

Compute the indefinite integrals. $$ \int 3^{x} d x $$

Compute the indefinite integrals. $$ \int 2^{x} d x $$

Compute the indefinite integrals. $$ \int 4^{-x} d x $$

Compute the indefinite integrals. $$ 3^{-2 x} d x $$

Compute the indefinite integrals. $$ \int\left(x^{2}+2^{x}\right) d x $$

Compute the indefinite integrals. $$ \int\left(x^{-3}+3^{-x}\right) d x $$

Compute the indefinite integrals. $$ \int\left(\sqrt{x}+\sqrt{e^{x}}\right) d x $$

Compute the indefinite integrals. $$ \int\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{e^{x}}}\right) d x $$

In Problems 97-122, evaluate the definite integrals. $$ \int_{1}^{4}(3+2 x) d x $$

Evaluate the definite integrals. $$ \int_{0}^{3}\left(2 x^{2}-1\right) d x $$

Evaluate the definite integrals. $$ \int_{0}^{1}\left(x^{3}-x^{1 / 3}\right) d x $$

Evaluate the definite integrals. $$ \int_{1}^{2} x^{5 / 2} d x $$

Evaluate the definite integrals. $$ \int_{1}^{8} x^{-2 / 3} d x $$

Evaluate the definite integrals. $$ \int_{1}^{9} \frac{1+\sqrt{x}}{\sqrt{x^{3}}} d x $$

Evaluate the definite integrals. $$ \int_{0}^{2} t(t+3) d t $$

Evaluate the definite integrals. $$ \int_{0}^{2}(2+3 t)^{2} d t $$

Evaluate the definite integrals. $$ \int_{0}^{\pi / 4} \sin (2 x) d x $$

Evaluate the definite integrals. $$ \int_{-\pi / 3}^{\pi / 3} 2 \cos \left(\frac{x}{2}\right) d x $$

Evaluate the definite integrals. $$ \int_{0}^{\pi / 4} \sin \left(x-\frac{\pi}{4}\right) d x $$

Evaluate the definite integrals. $$ \int_{0}^{1} \sin (\pi(x+1)) d x $$

Evaluate the definite integrals. $$ \int_{0}^{1} \frac{1}{1+x^{2}} d x $$

Evaluate the definite integrals. $$ \int_{-1}^{1} \frac{4}{1+x^{2}} d x $$

Evaluate the definite integrals. $$ \int_{0}^{1 / 2} \frac{1}{\sqrt{1-x^{2}}} d x $$

Evaluate the definite integrals. $$ \int_{-1 / 2}^{1 / 2} \frac{2}{\sqrt{1-x^{2}}} d x $$

Evaluate the definite integrals. $$ \int_{0}^{\pi / 2} \cos 2 x d x $$

Evaluate the definite integrals. $$ \int_{0}^{\pi} \sin 2 x d x $$

Evaluate the definite integrals. $$ \int_{-1}^{0} e^{3 x} d x $$

Evaluate the definite integrals. $$ \int_{0}^{\pi / 2} x^{2} \exp \left(x^{3}\right) d x $$

Evaluate the definite integrals. $$ \int_{0}^{\pi / 4} \sec ^{2} x d x $$

Evaluate the definite integrals. $$ \int_{1}^{e} \frac{1}{x} d x $$

Evaluate the definite integrals. $$ \int_{0}^{1} \frac{1}{z+1} d z $$

Evaluate the definite integrals. $$ \int_{0}^{1} \frac{1}{1+2 u} d u $$

Evaluate the definite integrals. $$ \int_{1}^{2} \frac{2}{3 t+1} d t $$

Suppose that $$ \int_{0}^{x} f(t) d t=2 x^{2} $$ Find \(f(x)\).

Suppose that $$ \int_{0}^{x} f(t) d t=e^{x}-1 $$ Find \(f(x)\).

Suppose \(\int_{\pi}^{x} f(t) d t=\sin x+C\) for some constant \(C .\) Find the function \(f(x)\) and the constant \(C\).

Suppose \(\int_{0}^{x} f(t) d t=\frac{1}{x^{2}+1}+C\) for some constant \(C\). Find the function \(f(x)\) and the constant \(C\).