91Ó°ÊÓ

Derive the given formulas. $$\int x^{n} e^{x} d x=x^{n} e^{x}-n \int x^{n-1} e^{x} d x$$

Use the Fundamental Theorem to calculate the definite integrals. $$\int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x$$

Derive the given formulas. $$\int x^{n} \cos a x d x=\frac{1}{a} x^{n} \sin a x-\frac{n}{a} \int x^{n-1} \sin a x d x$$

Evaluate the indefinite integral, using a trigonometric substitution and a triangle to express the answer in terms of \(x .\) Assume \(-\pi / 2 \leq \theta \leq \pi / 2\) $$\int \frac{1}{\left(25+4 x^{2}\right)^{3 / 2}} d x$$

Use the Fundamental Theorem to calculate the definite integrals. $$\int_{-1}^{e-2} \frac{1}{t+2} d t$$

Derive the given formulas. $$\int x^{n} \sin a x d x=-\frac{1}{a} x^{n} \cos a x+\frac{n}{a} \int x^{n-1} \cos a x d x$$

Use the Fundamental Theorem to calculate the definite integrals. $$\int_{1}^{4} \frac{\cos \sqrt{x}}{\sqrt{x}} d x$$

Evaluate the indefinite integral, using a trigonometric substitution and a triangle to express the answer in terms of \(x .\) Assume \(-\pi / 2 \leq \theta \leq \pi / 2\) $$\int \frac{1}{\left(16-x^{2}\right)^{3 / 2}} d x$$

Evaluate the indefinite integral, using a trigonometric substitution and a triangle to express the answer in terms of \(x .\) Assume \(-\pi / 2 \leq \theta \leq \pi / 2\) $$\int \frac{x^{2}}{\left(1+9 x^{2}\right)^{3 / 2}} d x$$

Use the Fundamental Theorem to calculate the definite integrals. $$\int_{0}^{2} \frac{x}{\left(1+x^{2}\right)^{2}} d x$$